«H -—'<-Q- ~'o>.. J-‘AA 4 _ .c 1r y'_- « «u -. ‘- d “7‘ ra’n.::v .17.: 9‘1 5::0‘. .‘ o- 6.3 3-; 3 ‘l‘ u .r 541 V’ ‘ x 5.4"! .. (T L;.. i "‘81 '_| l -5 a. - a“ J -711 _. - M“ I; , gear. {3 if. ‘ 2:- 72.‘ r -. ,_ ‘4 .4 ‘3‘ ' ‘3: H' raga . ‘73 395? 3.1 r 4?" ' $3143. A ’fzfi‘ a: . d2 .3 ‘ 1’1 TH ESts ‘ \ 2 i r-.. ... a i L? eh" 4' m" .- :3»de Viigt‘éf . wiuite ; ,,...j"'_ .- .,:‘_,1.~,_, ‘ U355 ‘ W. 4; «1“. L937 ‘ This is to certify that the dissertation entitled Stability and Nonlinear Response of Deck-type Arch Bridges presented by Khaled Yagoob Medallah ‘, f has been accepted towards fulfillment ofthe requirements for Ph . D . degree in Civil Engineering «Hf/me. Major professor Date March 26, 1984 MSUi: an Affirmative Actionv.~’Equal Opportunity Institution 0-12771 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 10692 4081 IV1f31_J RETURNING MATERIALS: Piace in book drop to LIBRARJES remove this checkout from w your record. FINES wiII be charged if book is returned after the date stamped beIow. STABILITY AND NONLINEAR RESPONSE OF DECK-TYPE ARCH BRIDGES BY Khaled Yagoob Medallah A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil and Sanitary Engineering 1984 ABSTRACT STABILITY AND NONLINEAR RESPONSE OF DECK-TYPE ARCH BRIDGES BY Khaled Yagoob Medallah The nonlinar response in three dimensions of deck-type bridges was considered by an “amplification factor method” which has a form similar to that used in the design of beam-columns. For use in the amplification factor method the eigenvalues (and corresponding eigenvectors) or buckling loads of the structures were studied. Two three-dimensional numerical model bridges were constructed from actual designed bridges. The . finite element formulation was used in the analysis using stright beam and truss elements; only geometric nonlinearity was considered. Factors affecting the elastic stability in three dimensional space studied included the patterns and the amount of rib bracing, the deck-ribs connections, and the stiffnesses of the towers. deck, and the transverse bracing. Predictions of nonlinear responses using the amplification factor method were compared with nonlinear equilibrium solutions for lateral. longitudinal, and vertical loadings. The comparisons in general indicated good agreement. ACKNOWLEDGMENTS The author would like to express his sincere appreciation to those who made the completion of this research possible. Special appreciation is extended to my dissertation advisor. Dr. Robert K. Wen, for suggesting this topic and for his continuous advice and encouragement throughout the work. Thanks and gratitude are also due to the other members of the Guidance Committee: Dr. William Bradley, Dr. Charles Cutts and Dr. Nicholas Altiero. The support and encouragement of Dr. William Taylor, Chairman of the Department of Civil and Sanitary Engineering. are sincerely appreciated. Also, thanks are due to the Saudi Arabian Educational Mission for supporting the research and to Vicki Switzer and Debbie Lange for their dedicated typing. LIST OF TABLES TABLE OF CONTENTS LIST OF FIGURES CHAPTER I. II. III. INTRODUCTION 1.1 Object and Scope. 1.2 Literature Review. 1.3 Nomenclature THEORETICAL BACKGROUND NMNM fiUNI—I 2.5 Introduction. The Amplification Factor. Equilibrium and Eigenvalue Equations. Eigenvalue Solution. 2.4.1 Introduction. 2.4.2 Solution Procedure for Eigenpairs. Tilted Load Effects. BRIDGES, MODELLING. COMPUTER PROGRAMS AND VALIDATION (0030 (DMD-D (.00) 0* Introduction. The PHAB. The MCSCB Bridge and Modelling. 3.3.1 Introduction. 3.3.2 Rib Cross-Sectional Properties. 3.3.3 Modelling of the Deck. 3.3.4 Modelling of Rib Bracing. 3.3.5 Modelling of Tower and Columns. Computer Programs. Validation. 3.5.1 Introduction. 3.5.2, Vertical Stability. 3.5 3 Lateral Stability. 3.5.3.1 Equation 16.25 in Reference 10. 2 Ostlund's Data. 3 The Data of Tokarz and Almeida. Truss Elements. Tolerance Effect. 3.5.5.1 Equilibrium Solutions. 3. .3. 3.5.3. (.00) mo- 0% iii PAGE vii 15 15 15 18 23 23 24 25 27 27 28 28 28 29 30 32 3d 35 36 36 36 37 38 38 39 4O 41 ‘1 3.5.5.2 Eigenvalue Solutions. IV. BUCKLING LOADS AND MODES 4.1 Introduction. 4.2 Bracing between Ribs - Bracing Patterns. 4.3 Bracing between Ribs - Amount of Bracing. 4.3.1 MCSCB Ribs (with X-truss Bracing). 4.3.2 FHAB Bridge (with Beam Bracing). 4.4 In-Plane Effect of Deck. 4.5 Effects of Type of Deck-Rib Connections on In-Plane Buckling. 4.6 Effects of Towers, Deck, and Transverse Bracing on Lateral Buckling. V. NONLINEAR RESPONSES 5.1 Introduction. 5.2 Lateral Response. 5.2.1 Nonlinear Equilibrium Solution. 5.2.2 Maximum Lateral Response by AP Method. 5.3 Longitudinal Response. 5.3.1 Nonlinear Equilibrium Solution. 5.3.2 Maximum Longitudinal Response by AP Method. 5.4 Vertical Response. 5.4.1 General. 5.4.2 All Columns Pinned. 5.4 3 Crown Column Rigidly Connected, All Other Columns Pinned. 5.4.3.1 Uniform Loading. 5.4.3.2 Nonuniform Loading. 5.4.4 Ribs and Deck Rigidly Connected. VI. SUMMARY AND CONCLUSIONS 6.1 Summary. 6.1.1 General. 6.1.2 Buckling Loads and Modes. 6.1.3 Accuracy of the Amplification Factor Method. 6.2 Concluding Remarks. TABLES FIGURES BIBLIOGRAPHY APPENDIX A APPENDIX B iv 42 43 43 44 47 48 49 5O 52 54 58 58 58 58 59 6O 6O 60 61 61 61 62 62 62 63 64 64 64 65 66 67 69 86 154 157 158 TABLE 3-1 LIST OF TABLES Cross-Sectional Properties of MCSCB, Four Panels, with Deck. Cross-Sectional Properties of MCSCB, Eight Panels. Values of Critical Loads for Lateral Buckling (kips/ft). Cross-Sectional Properties for Ostlund's Arches. Comparison with Ostlund's Data. Comparison with Data of Tokarz and Almeida. Results for Cantilever Column and Tower. Tolerance Effect on Eigenvalue Solutions. Buckling Loads for Different Rib Bracing Patterns. Effect of Bracing Cross-Sectional Properties on Buckling Load, FHAB Bridge. Tilted Load and Deck Stiffness Effect. Effect of Column Connections on In-Plane Buckling. Effect of Column Heights on In-Plane Buckling. Tower, Deck, and Transverse Bracing Effect on the Bridge Stability. Lateral Nonlinear Responses by Equilibrium and Amplification Factor Method. Longitudinal Nonlinear Responses by Equilibrium and Amplification Factor Method. Vertical Nonlinear Responses by Equilibrium and Amplification Factor Method - "All Columns Pinned.“ PAGE 69 69 7O 7O 71 72 72 73 74 75 76 78 79 8O .81 82 Vertical Nonlinear Responses by Equilibrium and Amplification Factor Method — "Crown Column Rigidly Connected, All Others Pinned, w Uniform." F 83 Vertical Nonlinear Responses by Equilibrium and Amplification Factor Method - "Crown Column Rigidly Connected, All Others Pinned, Nonuniform Loading." 84 Vertical Nonlinear Responses by Equilibrium and Amplification Factor Method - "Ribs and Deck Rigidly Connected." 85 vi LIST OF FIGURES Types of Arch Bridges. Beam Subjected to Combined Axial and Lateral Load. Coordinate Systems for Arch Bridges. Loading Conditions. Typical Buckling Modes. End Displacements of Three Dimensional Beam Element. Cross-Section of Beam Element. Tilted Load Effect. Overall Dimension of FHAB Bridge. Cross-Sectional and Material Properties of FHAB Bridge. Model for MCSCB Bridge. 4 and 8 Panel Models for MCSCB Bridge. Local x and z Coordinates for Truss and Beam Members. A Three Cell Box Model for Torsional Stiffness of Deck Beams. Components of Actual Bridge Deck. Modelling of Deck Slab.. Modelling of K-Bracing between Ribs. Sketch of Actual Tower. Equilibrium and Eigenvalue Solution of FHAB. Dimensions and Properties of Arch—A. vii PAGE 86 87 88 89 9O 91 91 92 93 94 95 96 97 97 98 .99 100 101 '102 103 3-13 3—14, 3-15 3-16 4-10 4-11 4-14 4-15 Equilibrium and Eigenvalue Solution of Arch-A. Arch Bridges Considered by Ostlund. Properties of Truss and Truss-Beam Towers. Buckled Shape of a Beam-Truss Tower and a Truss Tower. Pattern Buckled Bracing Buckled Bracing Buckled Bracing Buckled of Rib Bracing. Shape of MCSCB (First Mode). Shape of MCSCB (Second Mode). Shape of MCSCB (Third Mode). Shape of MCSCB (First Mode)- Buckled (Second Buckled Shape of MCSCB Mode). Shape of MCSCB (Third Mode). Buckled (Fourth Buckled Bracing Buckled Bracing Buckled Bracing Buckled Bracing Buckled Bracing Shape of MCSCB Mode). Shape of MCSCB (First Mode). Shape of MCSCB (Second Mode). Shape of MCSCB (Third Mode). Shape of MCSCB (Fourth Mode). Shape of MCSCB (Fifth Mode). Ribs, Ribs, Ribs, Ribs, Ribs, Ribs, Ribs, Ribs, Ribs, Ribs, Ribs, Ribs, X-Truss X-Truss x-Truss D-Bracing D-Bracing D-Bracing D-Bracing Transverse- Transverse- Transverse- Transverse- Transverse- Effect of Amount of Bracing on Lateral Buckling Load (Ab _ 1/5 Rib Area). 0 Buckled Shape of the MCSCB, 8 Panels, ”0 DOOR: Ab/Ab . .001. O . viii 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 4-19 4320 4-21 4-22 4-23 4-24 4-25 4-26 4-27 4-28 4-29 4-30 ".—31 4-32 4-33 Buckled Shape of the MCSCB, 8 Panels, No Deck, Ab/Ab = O 01. o Buckled Shape of the MCSCB, 8 Panels, No Deck, Ab/Ab , 0.25. o Buckled Shape of the MCSCB, 8 Panels, No Deck, Ab/Ab0 3 0.5. Buckled Shape of the MCSCB, 8 Panels, No Deck, Ab/Ab s 1,0, 0 Buckled Shape of the MCSCB, 8 Panels, No Deck, Ab/Ab a 2.0. o The Lowest Anti-Symmetric Out-of-Plane Mode for x-Truss Braced Bridge. Different Types of Column Connections between Deck and Arch Ribs. Effect of Deck Elevation and Different Deck-Ribs Connections. Buckled Shape of MCSCB, Single Rib with Deck, Pinned Columns. Buckled Shape of MCSCB, Single Rib with Deck, Column at Crown is Rigidly Connected (EIsEIo) Buckled Shape of MCSCB, Single Rib with Deck, Column at Crown is Rigidly Connected (EI=10EIO) Buckled Shape of MCSCB, Single Rib with Deck, All Columns are Rigidly Connected (EI=EI°) Buckled Shape of MCSCB, Single Rib with Deck, with Spandrel Bracing in One Panel. Buckled Shape of MCSCB, Single Rib with Deck, with Short Pinned Column at Crown. Buckled Shape of MCSCB, Single Rib with Deck, Ribs and Deck Rigidly Connected. Buckled Shape of MCSCB, 4 Panels, No Transverse Bracing (First Mode). Buckled Shape of MCSCB, 4 Panels, No Transverse Bracing (Second Mode). Buckled Shape of MCSCB, 4 Panels, No Transverse Bracing (Fourth Mode). ix 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 4-34 .435 4-36 4-37 4-38 4-39 Buckled Shape of MCSCB, 4 Panels, No Transverse Bracing (Fifth Mode). Buckled Shape of MCSCB, 4 Panels, Transversely Braced at Crown (First Mode). Buckled Shape of MCSCB, 4 Panels, Transversely Braced at Crown (Third Mode). Buckled Shape of MCSCB, 4 Panels, Uniformly Transversely Braced (Third Mode). Buckled Shape of MCSCB, 4 Panels, Uniformly Transversely Braced (Fifth Mode). Buckled Shape of MCSCB, 4 Panels, No Transverse Bracing, Deck Stiffness Reduced by 10 (First Mode). Buckled Shape of MCSCB, 4 Panels, No Transverse Bracing, Tower Stiffness Reduced by 10 (First Mode). Effect of Deck and Tower on Out-of—Plane Buckling. Equilibrium Solutions of Vertical and Lateral Displacement at Crown. Solutions of Lateral Displacements for Three Points on Bridge. Equilibrium Solutions of Vertical and Longitudinal Displacement at Crown. Loading Conditions for Symmetric and Anti-Symmetric In-Plane Buckling. Comparison of Vertical Responses for Nonuniform Loading. 141 142 143 144 145 146 147 148 149 150 151 152 153 CHAPTER I INTRODUCTION 1.1 Object and Scope. Arch bridges are known for their aesthetical lines and the large distances they can span. They are usually built of concrete or steel to overpass rivers, to connect roads over valleys in mountain areas, to bridge highways, and sometimes even to support special structures. According to the deck location arch bridges are classified as "deck bridge," ”half-through bridge” or "through bridge.” Figure 1-1 illustrates the different types of arch bridges. An arch bridge subjected to a vertical uniformly distributed load would develop mainlyv thrust actions, a situation similar to that of a column under axial compression. Dead loads are essentially uniformly distributed vertical loads. Live loads, on the other hand, may cover only a portion of the span to produce maximum bending. Wind load (and earthquake loads) are applied laterally or longitudinally. Due to the initial compression in the arch in response to the dead loads, the structure behavior ‘under ‘additional- live load '(or wind load) will be nonlinear. m. is a situation similar to the case of a beam subjected to an axial load and lateral load. Starting about the late forties, with increasing needs I‘d for more economical and slender arches the deflection effect, or the geometric nonlinearity effect on their response became an important issue. Many attempts have been made to address the problem. While design engineers used relatively simple approaches, researchers studied it in theoretically more rigorous frameworks. (See section on ”Literature Review.') Today, a fully nonlinear solution is possible with the help of finite element codes and advanced computing facilities. Such a solution is very expensive and not yet readily available to many designers. A less involved and simpler approach is desirable to facilitate, at least, preliminary designs. A procedure that had been used to estimate nonlinear response, Rn, from the linear response. RL' is as follows: R = 1 (11) n RL 1 _ I: ' P c where P is the compression load on a structure and Pc is the critical buckling load of the same structure. Equation (1.1) may be written as: 3a" .31. AF. in which AF (stands for “amplification factor") - 1/(1-P/Pc). This procedure will be referred to in this thesis as the “amplification factor method.” The major objective of this work is to examine the validity of the amplification factor method for estimating nonlinear responses of arch bridges. ‘ Since AF involves Pc' obviously the buckling load is of great importance in this work. In the literature the effects of some factors such as the torsional rigidity of the ribs and the flexural rigidity of the bracing members on the buckling load have been reported. It seems that there are other factors such as the deck rigidity and the bracing which may be important factors also. The effect of such factors on the buckling loads of arch bridges will also be considered here. The study was based on a nonlinear elastic beam finite element as reported in Reference 23. The computer program used for this study was obtained by modifications and expansion of two available programs CURVEL and FRALSD (21,22). The modifications comprised of the treatment of larger systems including the loading vectors, the addition of a nonlinear elastic truss element, solution for multiple eigenpairs (eigenvalues and eigenvectors), and the graphic output of the buckling modes. The computer program was extensively checked by comparing results with known correct ones. Two -bridge models were used for this research. .One. designated as MCSCB, was a modified version of the Cold Spring Canyon Bridge in California (20), and the second, designated as FEAB, was taken from a Federal Highway Administration Report on arch bridges (11). The use of 4 such model structures of practical design should increase the significance of the data and the value of the results. The most important finding of this study is perhaps that, so far as elastic stability is concerned, the arch bridge should be considered as a system. In the past, much attention had been focussed on the stability of arch ribs. The results of the study showed that other components such as the deck, longitudinal bracing, transverse bracing, as well as rib bracing, had very significant effects on the stability of the bridge. For example, an addition of nominal transverse bracing members at the crown could change the buckling mode and correspondingly significantly increase the buckling load. The amplification factor method was considered for lateral, longitudinal and vertical loadings. The values of the maximum responses compared well with those obtained from solutions of the nonlinear equilibrium equations if the loads are, say, no greater than one half of the buckling load. In regard to buckling load, the value used in Eq. 1.1 must be that which corresponds to a buckling mode compatible to the deflection shape caused by the applied load. 1.2 Literature Review. The problem of out-of-plane buckling of a curved beam was studied by Ojalvo and Newman (1). The authors presented linearized perturbation equations ‘ about a reference state, which satisfy the nonlinear beam 5 equilibrium equations. Equations for the determination of the buckling load could be solved by a "shooting" procedure (a boundary value problem solved as an initial value problem). Ojalvo, Demuts and Tokarz (2) presented two equations for determining the out-of-plane buckling of a planar curved member initially in a plane under pull and thrust, with the load also initially in plane. Wen and .Lange (3,21) presented a curved beam finite element model including geometric nonlinear effects. The element is initially curved in a plane but may deform out of or in the initial plane. Members of various shapes can be studied due to the flexibility of the geometry of the finite element model. Presented data for in-plane and out-of—plane buckling showed that for symmetric and symmetrically loaded arches using the linear or quadratic eigenvalue formulations made only insignificant differences in the buckling loads. Two matrices were considered for the first nonlinear stiffness, one with rotation terms included and one without. Results showed that the two agree for problems with little bending, but for problems involving significant bending the latter should be used. Tokarz (4) presented experimental results on the lateral .buckling of parabolic. arches.. Tokarz and Sundhu (5) presented a pair of equations governing the torsional buckling of arches under uniform vertical loading based on linear buckling theory. Both papers used a free standing rib, two ribs braced at the crown, and used uniform bracing only in the experimental work (4). Factors affecting the torsional buckling such as the rise to span ratio, the torsional rigidity to the out-of—plane bending rigidity of the arch, and "tilted load effects" (see section 2.5) were considered. Some of the tabulated results will be compared with solutions obtained by the finite element model used herein. Donald and Godden (6) presented a solution for the problem of a curved beam with transverse loading only and transverse with axial loading using a numerical forward integration method (a "shooting" procedure). Results correlated well with experimental work. An amplification factor method related to the thrust in the arch gave predictions with accuracy to one percent. Only symmetric lateral buckling was studied. The problem of tied arches (for through bridges) was studied by Godden (7) and Godden and Thompson (8). Theoretical and experimental work were presented for various factors affecting the lateral stability of unbraced tied arch bridges, such as flexural rigidities, torsional rigidities, rise to span ratio and the stiffness of the hangers connected to the tie. The hanger effect was found to“ be _very significant. AA good. correlation. between theoretical and experimental results was noted. Shukla and Ojalvo (9) presented a numerical solution, based on a forward marching integration, for the theory developed in (1). It was found that the buckling load for through arches is three to four times that for deck type arbhes. Optimum rise to span ratio with torsional to flexural rigidities are presented. Only single rib arches were used in the study. In the book Guide to Stability Desigp Criteria of Metal Structures, third edition (10), were summarized available data in the form of formulas and tables for arch design for in-plane and out-of-plane elastic stability under different loading conditions. In Nettleton's work (11) related to the design of steel and concrete bridges, suggestions for the practitioners were presented regarding local and overall stability design. Moment magnification was related to the thrust at supports. Wind load design procedures were suggested for both the single and double lateral systems. Based on data in Reference (10), the effective slenderness ratios for the braced arches similar to the concept of column design were proposed for different rise to span ratios with the effective length given by Bleich (12). He also suggested that a continuous roadway would contribute to the stiffness of the arch in proportion to the ratio of the floor member depth to the depth of the rib. Ostlund. (13) was the_ first to study two ribs braced with cross beams. Two arch forms (one "deep,” one ”shallow”) with varying properties were studied. Factors considered included the flexural rigidity of transverse bars, rise of arch, spacing of the two ribs, number of transverse bars, torsional stiffness of ribs, the "tilted load" effects, and connection of ribs to the crown. A good correlation between the theoretical and experimental results were found by the author. The arches used had the out-of-plane mode as the lowest buckling mode. In practice it would appear that the in-plane buckling load would be the lowest. Wastlund (14) presented several works done by several researchers under his supervision. A method analogous to that of the. beam-column was suggested for arches with vertical buckling. A simple formula for additional moment is presented (amplification factor based on the arch thrust). A method for calculating the vertical symmetrical and antisymmetrical buckling load was also' discussed. Tests results. which correlated well with theoretical data were presented. For out-of-plane' buckling of braced arches, Wastlund suggested an approximate method which stated that arches should be straightened out so that the ribs and the bracing would lie in one plane. The buckling load then would be computed as for a column with battens. This method ignores the rise to span ratio, the torsional stiffness of the rib and the vertical stiffness of the bracing., Almeida (15) presented numerical solutions of the curved beam theory (2) for two parabolic ribs braced with tranverse beams. The cases of the deck situated above, below, and no decks were considered. Factors affecting.the buckling load were studied. Almeida's data checked with the work of Tokarz fairly well and with Ostlund satisfactorily. Sakimoto and Namita (16) used the transform matrix method to obtain the eigenvalues of the differential equation governing the buckling load of framed structures. Only circular arches under uniformly distributed radial forces were considered. The authors found that in certain cases the location of the transverse bars was more important than the number of them and that to increase the strength of the arch it was better to constrain the out-of-plane flexure rigidity of the arch than the torsional deformation, and that a slight loosening of the fixed end about the out-of-plane may reduce the out-of-plane buckling of the arch considerably. Sakimoto and Komatsu (17,18) studied the ultimate load-carrying capacity of arch systems under vertical loading. The papers considered the overall slenderness ratio of the structure as a major parameter. The second paper presented for through bridges gave three effective length values to be used under certain conditions for buckling load estimation. A small initial deflection was .assumed to- induce buckling.» A design. formula «for.the preliminary design of through bridges was suggested. Yabuki and Vinnakota (19) did a literature review that covered a very broad area of arch stability including 'planar and -spatial, linear, nonlinear (geometrical and 10 material nonlinearity) and some parameters affecting the stability problem. A look at the German and Japanese specifications with suggestions based on the presented data was also included. 1.3 Nomenclature. A A“ A, B I Cross-sectional area; I Cross-sectional area of a deck beam; I End nodes of an element: I Cross-sectional area of the modelled column (MCSCB): I Cross-sectional area of chord in original and modelled bridges: I Cross-sectional area of composite beam: I Cross-sectional area of D-truss bracing: I Cross-sectional area of diagonal member in original and modelled bridges; I Cross-sectional area of spandrel bracing: I Cross-sectional area of deck slab: I Cross-sectional area of transverse bracing between ribs (beam bracing): I Cross-sectional area of truss member; I Cross-sectional area of vertical member in original and modelled bridges; I Chord length in truss (CSCB): I Parameters used.for definition of shape functions: I Modulus of elasticity: I Rise; I Shear modulus; I Neight'of crown column; KT [1!] , [K] [n1]. [n1] tn,:1. (81:1 Inf]. [N2‘I P {P} {Pref} 11 Original height of crown column as modelled for (MCSCB): Moment of inertia of column as used for (MSCSB); Moment of inertia due to beam action in beam-truss tower; Moment of inertia for beam-truss tower; Moment of inertia due to truss action in truss and beam-truss towers; Moment of inertia of cross-section (Figure 3-5); Moment of inertia of deck about global Y-Y axis; Torsional constant; Torsional constant of cross-section; Element and structural linear stiffness matrices: Arch bridge span: Length of element: Length of diagonal member in truss (original and modelled bridges); Length of vertical member in truss (original and modelled bridges): Total number of panels; Element and structural first order nonlinear stiffness matrices: Element and structural first order ,geometric stiffness matrices; Element and structural second order nonlinear stiffness matrices; Applied concentrated load; External load vector; Reference external load vector; 12 {PC} I Critical value of applied load; Pc I Critical (buckling) load; Q I Lateral concentrated load; (Q) I Structural generalized displacement vector; {00} I Unit vector; {01) I First eigenvector; {6°} - {no} - a, {01) {0,9,} I Reference structural generalized displacement vector; go I Axial compression at crown in rib; . . T, (q, [gll Q20 ' 0 0 QSI Q70 get 0 ' ' Q12] 0 I Response; Linear response; fr?" I Nonlinear response; S I Spacing between ribs or deck width; SD I Force in diagonal member in truss; Sv I Force in vertical member in truss; [8.] I Structural secant stiffness matrix; [8T] I Structural tangent stiffness matrix; u, v, w I Displacements along local x, y, z axes, respectively; “1' v1, w1 and I Displacement for nodes 1 and 2 of beam 'uz, v2, w2 element along x, y, z axes, respectively; *2 " s:::.:fa:::r:2::.ra.éut'°‘ ’4'“ Us I Strain energy of the element; Ut I Torsional strain energy of the element; U I Total strain energy of the'elemsnt; "2' "3' "4 ¢. w. 6 $10 $1: 61 and ¢2' W2: 62 ¢p A A A8, A8* AB' AB' 6. 6'I l3 Quadratic, cubic and quartic parts of strain energy; Total volume of bracing members between ribs; Nonuniform distributed load; Additional applied load (distributed on portion of the span to produce maximum response); Critical uniform load; Fixed load (uniformly distributed); Uniformly distributed yield load. wc/wY Global coordinates of the bridge (see Figure 2-2): Local coordinates of element (see Figure 3'5): "F/wc‘ Constant, see equation 2.4.9; Longitudinal strain; Rotation about x, y, z axes, respectively; Rotation about x, y, z axes for nodes 1 and 2, respectively; Total potential energy; Buckling load parameter; Incremental operator; Shear displacement in original and modelled bridge; Bending displacement in original and modelled bridge; Shear stress; Angle of inclination of diagonal truss member in original and modelled bridge; 14 = Coordinates of a point with respect to principal axes; = Yield stress. CHAPTER II THEORETICAL BACKGROUND 2.1 Introduction. In this chapter the basis of the amplification factor method is explained in detail. For the sake of completeness, an outline of the derivation of the equilibrium and eigenvalue equation (21,22) is presented next, followed by a description of the procedure used for the solution of the eigenvalues and eigenvectors. The chapter concludes with a description of the so-called “tilted load effects." 2.2. The "Amplification Factor Method". It was mentioned in the preceding chapter that the "Amplification Factor", AF, was introduced by Timoshenko (27) when considering a simple beam problem subjected to a combined lateral and axial load (see Figure 2-1). Using the principle of superposition a trigonometric series for the beam deflection was obtained.. The first term in the 'series for the mid-span deflection, R, is 3 11" BI 1.0: where Q is the lateral load, E1 is the flexural rigidity, i is the beam length, and a I P/P in which P is the of applied axial load on the beam, and Pc is its lowest 15 16 buckling or critical value. The value of the linear deflection, RL' is approximately equal to the first factor in equation 2.2.1, i.e., 023 ,1 :49 23 =figér (2.2.2) Thus equation 2.2.1 may be rewritten as: i 1 R " RL (Fa) (2.2.3) The term (1/1- a ), as mentioned earlier, is the amplification factor, AF. The preceding concept had been employed to estimate the nonlinear response of other types of structures. For example, for a single arch, the factor a was expressed as the ratio of the actual thrust and the critical value of the thrust for the arch (see, for example, (6)). For a three dimensional arch bridge it is more convenient to use the AF as a function of the applied loads. If the vertical fixed load (usually a uniformly distributed "dead load") is denoted by wF, and the additional load, "live or wind load" by w then the al nonlinear response, Rn, due to w may be estimated a by 17 _ 1 Rn "’ RLFE (2.2.4) where RL is the linear response due to wa and a = wF/wc. The quantity wc is the lowest "compatible buckling load." The preceding term is uSed here to define the buckling or critical value of the vertical load the associated buckling mode of which is conformable or compatible with the deflection shape of the bridge due to the added load under consideration, i.e., Wa. A space structure has an infinite number of We, each corresponding to a different buckling mode, and so care is required to use the correct wc for the AF method. The global coordinate system is presented in Figure 2-2. The combinations of loading conditions are presented in Figure 2-3, and two typical buckled shapes are presented in Figure 2-4. For response in the X or Y direction, generally WC that causes the buckled shape illustrated in Figure 2-4a is to be used, for response in the 2 direction, generally WC that causes the buckled shape illustrated in Figure 2-4b is to be used. It is very essential to note that depending on the structure-load system, the compatible buckling mode shape may be different from those given in Figure 2-4. This will be pointed out in Chapter V. 18 2.3 Equilibrium and Eigenvalue Equations. The nonlinear equilibrium solutions and the buckling loads and modes used herein are based on the nonlinear finite element model described in references 22 and 23. For the sake of completeness, the method is outlined in the following. The following assumptions were made for the derivation of the model: (1) The material of the elements is linearly elastic. (2) Plane sections remain plane. (3) The cross-section of the element is constant and has two axes of symmetry. (4) The effect of torsional deformation on normal strain is negligible. (5) The axial strain due to the transverse displacement is averaged over the element length. Consider a beam element in space with x, y and z - coordinates and u, v and w displacements,d»qiand 6 are rotations about x, y and z axes respectively. An assumed linear shape function for each of u and ¢ and a cubic shape function for each of v and w are: u=a1+a2x 2 3 v I a3 + a4x + a5x + 36x 2 - 3 w - a7 + aax + agx + 310x ¢ = a11 + a12x (2.3.1) Referring to Figure 2-5 the boundary conditions are: 19 at x = O _ _ _ 93L: u-ul, v—vl, w—wl, dx 61 (2.3.28) Q1. _ _ dx 1121 and (b - ¢1 and at x = 2 g _ - 92.: u 112, v — v2, w — w2, dx 62 (2.3.2123) Qt. - _ .. dx - (p2 and q) - ¢2 Substituting equation (2.3.1) into equation (2.3.2), a system of linear equations would be obtained. When solved, the values of u, v, w and ¢ as functions of the generalized coordinates are found. Using beam theory where plane sections remain plane, the longitudinal strain of beam elements is: dzv d2w €(x.n.C) = e (x) +n— +c— (2.3.3) a dx2 dx2 in which I] and c are the coordinates of the point from the cross-section centroid at which the strain is evaluated (see Figure 2'6). and ea(x) is the axial strain at the centroid which, when using the average strain assumption, is du 1 Il1 dv 2 ca(x) . “—3” '2— Cf’a-e—a dx (2.3.4) 1 £1 dw 2 From equation (2.3.3) and (2.3.4) the strain at each 20 point of the section can be determined readily. The strain energy due to normal strain is, UE = f i—E [E(X.n,C)2] dvol (2.3.5a) vol 2 and due to torsion 1 2' 2 u = e. f 63 (999 dx (2.3.5b) t 2 0 dx upon substitution “of equations (2.3.3) and (2.3.4) in equation (2.3.5a), and 553% = BEE; in equation (2.3.5b) then the strain energy. U, can be found: (2.3.6) The stiffness matrices can be derived from the strain energy equation (2.3.6). Equation (2.3.6) can be divided into three parts, i.e.; U = U2 + U3 + U4 (2.3.7) U2 contains only quadratic terms, and U3 in which, and U4 contain cubic and quartic terms respectively. The stiffness matrices can be derived as follows: azu2 3 (2.3.8) [(n1)1'31=['3§-1—3§;I 320, [(n2)i'jl = [W] 21 in WhiCh Q1 and q1 represent the generalized coordinates such as ul, v1, "1' ..., etc., [k] is the linear stiffness matrix, [n1] and [n2] are the first and second order nonlinear stiffness matrices, containing, respectively, linear and quadratic terms of the displacements. If terms containing rotational displacements in [n1] are eliminated, the resulting matrix is denoted by [n1*]. Details of the entries of the various stiffness matrices can be found in the above—cited references. The linear and nonlinear stiffness matrices for a truss member were developed in the same manner as for the beam. The matrices are listed in Appendix A. (A linear shape factor was assumed.) If the stiffness matrices of the elements are transformed into global coordinates, assembled, and denoted bY [K]: [N1] and [N2] for the linear, first and second order system stiffness matrices, and denoting the generalized displacement vector and external load vector by (Q) and (P), the total strain energy U and the potential energy Op can be written as follows (see Mallet and Marcal (25)): U = (ong— [K] + ,1;- [N1] + -§-§- [N21] (0) (2.3.9) and 22 ¢p = U - {Q}{P} (2.3.10) The first variation of the potential energy gives the equilibrium equation: [55] {Q} = (P) (2.3.11) where [S5] = [K] + %- [N1] + é- [N2] (2.3.12) The second variation of the potential equation gives the incremental equilibrium equation [ST] {AQ} = {AP} (2.3.13) where {130) and (AP) are the incremental displacement and loads respectively. [ST] is the tangent stiffness matrix which is given by: [ST] = [K] + [N1] + (N2) (2.3.14) from equation (2.3.14), the incremental equilibrium equation is (IKI + [N1] + [N21) (AQ) = (A2) (2.3.15) The nonlinear equilibrium responses obtained for this study were solutions of the preceding equation. The buckling loads and modes were obtained by setting (AP}={O) in the preceding equation, i.e., ([K] + [N1] + (1(2)) {AQ} = (0) (2.3.16) 23 The solution of equation (2.3.16) is considered in the next section. 2.4 Eigenvalue Solutipn. 2.4.; Introduction. The eigenvalue solution is a mathematical formulation used for the determination of critical loads. If a load on a structure is increased proportionally, the structure reaches a load called the buckling load at which the response could become indefinite. The solution of equation 2.3.16 is difficult due to the fact that the matrices [N1] and [N2] are functions of the displacement variables (q). Assuming that the displacements are linear functions of the applied loads up to the point of buckling, then: {Q = [kl-1 {P (2.4.1) ref} ref} where {Pref} is an arbitrary reference load vector and {Q ref} is the corresponding linear response. Since [N1] and [N2] are linear and quadratic functions of the displacements, writing (P) = 1 (Pre,) (2.4.2) one obtains ("1({Q})I = [N1({Qre,))]1 (2.4.3) and 24 [N2(IQ})I = (N2((Q,,,)))2 (2.4.4) in which A is a scaler parameter. Equation 2.3.16 can then be rewritten as 2 _ (IX) + Actull + K c [N2]){Qref} {A0} - 0 (2.4.5) Equation (2.4.5) is a quadratic eigenvalue equation. For sufficiently small displacement, the [N2] matrix can be neglected, i.e., ([x1 + lc[N1])(Q f} (A0} = 0 (2.4.6) re Equation (2.4.6) is a linear eigenvalue equation. A solution of the eigenvalue problem defined by equations (2.4.5) or (2.4.6) yields the value of Ac, and consequently the critical load, (PC), may be found using equation (2.4.2), i.e., {PC} = Ac (pref) (2.4.7) 2.4.2 Solution Procedure for Eigenpairs. To solve for the first eigenpair (eigenvalue and eigenvector), the well-known method of "inverse vector iteration" was used using a unit starting vector, i.e., (00}? I [1 1 1 .... 1]. The next eigenpair was obtained by "deflating" the iteration vector or sweeping out from the latter the eigenvector just computed, i.e., (QC) = {QC} - altol) (2.4.8) in WhiCh (Q1) is the first eigenvector just found. 25 If one writes: 00 = 41 Q1 + 32 02 + a3 03 + ... (2.4.9) the value Of 01 may be found using the orthogonality of the Q1 vectors with respect to [-N1], i.e.; {Qn} [-N1] (on) = 1 n s m I O n I m It follows: a, = (0,)Tt-N1)(Qo) (2.4.10) Iteration starting With {60) would produce a second eigenpair. Similarly, a third eigenpair can be obtained after sweeping the two eigenvectors from the unit vector and. use it for starting the iterative procedure. The details may be found in such works as reference 24. 2.5 Tilted Load Effects. In earlier research works on the effect of the bridge deck on the lateral buckling load, the deck was assumed to be rigid in that direction. Therefore, the deck vertical loads transferred to the arch ribs would be tilted on account of the lateral displacement. The phenomenon, illustrated in Figure 2-7, is similar to the "P-A" effect considered in building structures. It will also be referred to herein as "rigid or classical deck effect." For a finite element formulation, the tilted load effect had been considered in reference 21 for a single arch rib. 26 The same approach is used herein for the arch bridge system. CHAPTER III BRIDGES, MODELLING, COMPUTER PROGRAMS AND VALIDATION 3.1 Introduction. It was mentioned earlier that two bridges were used for this study: the FHAB and the MCSCB. The FHAB was taken from a research report by Nettleton (11) and used for. this research essentially as it was given therein. The MCSCB was obtained by modifying or simplifying the model for the Cold Spring Canyon Bridge. Two reasons make the simplification necessary. One is the computer solution cost. The other is the central computer memory. This analysis used 340,000 words, the full memory of the MSU CYB 750 has 377,000 words. Thus, the present study with the simplifying modelling already used about ninety percent of the central memory. The theoretical basis for the modelling is presented in section 3.3. Two computer programs were used for this study. Program NEAMAH and EIGGRAPH. Program NEAMAH is for the splutigns of linear and nonlinear equilibrium problems and‘ eigenvaluehmproblems; program EIGGRAPH is for the plotting of the buckled shapes. Many examples have been solved and some compared with available data in the literature in order to check and validate the program and the numerical procedures. The sensitivity of the eigenvalue solutions to the tolerance used in the numerical procedure was also 27 28 studied. 3.2 The PHAB. The FHAB bridge consists of two braced parabolic ribs (without a deck). The structure has thirteen panels with 28 nodes and 38 straight beam elements. The two ribs are supported by four hinges where no translations are allowed and only rotation about the transverse axis is permitted. The details of the structure are presented in Figures 3-1 and 3-2. The Figures include the cross-sectional area A in sq. ft., the moment of inertia about the horizontal (major principal) axis, Ixx' the minor principal axis, 122' and the torsional constant, KT, all in ft‘ (see Figure 3-5 for local coordinates). The transverse beam bracing between the two ribs is so oriented that the slope of its major principal axis is the average of the slopes of the two adjacent ribs (see reference 28. PP.. 288-291). 3.3 The MCSCB Bridge and Modelling. 3.3.1 Introduction. The Cold Spring Canyon Bridge (reference 20, pp. 13-22) is a deck-bridge located about 13.5 miles north of city limit of Santa Barbara, California. The bridge has eleven panels with a 700 ft. span , and 119 ft. rise. The ribs are spaced at 26 ft. apart. The original bridge is not symmetric with respect to the crown. The model MCSCB is symmetric with the following overall dimensions and 29 material properties: span I 700 ft; rise I 121.25 ft spacing between the two ribs I 26 ft deck elevation I 133.50 ft; deck width I 28 ft modulus of elasticity E I 0.4175 x 107 ksf Poisson's ratio I 0.3 A complete three dimensional model of the bridge must include the two ribs, the rib bracing, the deck, and its bracing, the columns, the towers, and sometimes, the spandrel and transverse bracings. A large number of elements are involved. In this case it has been practical to use only four panels for the bridge system. For the MCSCB four panel model, the number of elements is 54, the total number of nodes is 22, the total number of degrees of freedom is 90, and the bandwidth is 33. Eight panels can be and were used when the deck was eliminated or only the in:plane behavior of the bridge with the deck was considered. 3.3.2; Rip Cross-Secgggnal Properties. For four and eight panels the ribs are illustrated in Figure 3-4 and tabulated in Tables 3-1 and 3-2. In the following the theoretical basis for the modelling of the different components of the MCSCB is presented. 30 3.3.3 Modelling of the Deck. The deck was modelled by two beams braced with x-truss bracing (see Figure 3-3). The actual deck is mainly built of a slab carried by fgur stringers (see Figure 3-7). The moment of inertia for each beam in the model about x—x is calculated from simple statics for two composite stringers in the actual deck. For the torsional rigidity of the two beams, a three cell box is modelled to represent the four stringers, the deck slab, and the bottom lacing (see Figure 3-6). The torsional rigidity for each beam is taken to be 1/2 of that of the three cell model. The moment of inertia about the z-z axis for the modelled beam representing the deck and the cross-sectional area is calculated to provide a moment of inertia equal to that of the actual deck about the' global Y axis. The latter moment of inertia of the actual deck, I Y-Y’ which is illustrated in Figure 3-7b, may be obtained as 2 2 IY_Y = 2.ACB ((5/2) + (8/6) ) (3.3.1) where ACB is the cross-sectional area of the composite beam and S is the spacing between the two exterior stringers. Since IY-Y is to be provided by the two beams in the model, then IY—Y is equal to; .. E. IY-Y _ 2(Iz__z +.4 A ) (3.3.2) where Iz-z is the moment of inertia about the z-z local axis parallel to the global Y axis. Taking the latter 31 to be twice the local moment of inertia about the local z-z axis of a composite beam, then A‘, the cross-sectional area of a model deck beam, can be calculated readily. The deck slab can be modelled by an x-truss bracing to provide the same shear stiffness as that of the slab (see Figure 3-8). Denote the shear displacement of the slab in Figure 3-8a by As, then I> ll shearing strain - 1 (3.3.3) As ll 0| a )0 where 'r is the shear stress and G is the shear modulus, or, for a unit load, A .. 1 x 1 (3.3.4) slab G The x-truss bracing shear displacement is I 2 A I 2 SD id (3.3.5) 8 A* E d where SD is the force in the diagonal member, 1d is the length of the diagonal member, A"d is the diagonal member cross-sectional area, and E is the modulus of elasticity. Substitute for each value in equation 3.3.5. 32 (12+32)1/2]2 (12+82)1/2 x d A* = [1/2 2 (3.3.6) 5 The bracing and slab deck are to have the same shear displacement for the same (unit) shear load. Hence, AS = A*S ' (3.3.7) Substituting for As and A*s in 3.3.7 and simplifying, one obtains: * A d = G 1 2 -[(—) + -1——] (3.3.8) Aslab 2E S ( ) i 5 3.3.4 Modelling of Rib Bracing. The K-truss bracing for a panel between the two ribs of the real bridge is shown in Figure 3-9a. The x-truss bracing that is to replace the K—truss bracing is illustrated in Figure 3-9b. Under a unit shear load, the two trusses are to have equal displacements (sum of bending displacement AB or A*B and shear displacements As or Ms), i.e. If Sc is the force in the chord members, 1C is the member's length, E is the modulus of elasticity, and AC is the chord cross-sectional area. 33 A = 2 Z c C n = 1.2.3,...N (3.3.10) where N stands for the total number of panels - i.e. A = n 2 a COT26 “=1 (2) -E-X;—'2 (3.3.11) 2 a cor2e N n 2 z E Ac = where» 6 is the angle of the inclination for the diagonal as illustrated in Figure 3-9a; the chord length is a. ‘For the single panel x-truss: A* = Na corze' 2 A‘ E C (3.3.12) in WhiCh Atc is the chord area and 6* is illustrated in Figure 3-9b. The shear displacement of the K-truss can be represented by the following: 2 2 3 Ad E AVE ) N (3.3.13) where the subscript d stands for the diagaonal member and v for the vertical, if S is the spacing between the two ribs. Substitute for each value in equation 3.3.13, i.e.: A _ [2 (1/2 csce)2£ 2 (1/2)2 8/2 8 . Ad E d + Av E ] N (3.3.14) and similarly for the x-truss bracing, i.e. 2 2* 21* = (1/2 csce*)__;‘__ 2 (3.3.15) S A‘ E d where; 1/2 4*d = (N2a2 + 32) . id = (a2 + (S/2)2)1/2 2. gs csce = _Q_, csce* = __d 8/2 Na Setting AZIAc and substituting in equation 3.3.9. the value Of Atd can be readily found. A test problem involving modelling of the K-truss by x-truss was solved. The K-truss had four panels which was modelled by a one panel X-truss. The displacement of the free end of both cantilever trusses checked well which gives support to the validity of such modelling. 3.3.5 Modelling of Towers and Columns. The tower is, in actual practice, a pier in the form of a plane frame (see Figure 3-10). It is used to support the deck vertically, by transferring the loads to the foundation directly, and laterally, by providing a stiffness in the lateral direction. For the vertical stiffness of the tower, the tower is assumed to be perfectly rigid, which can be modelled by having a support in the Y-direction (see Figure 3-10). The lateral stiffness can be estimated by evaluating the actual stiffness of the frame which amounts to 1022 kips/ft. This is represented by a truss element with length 20 ft. and 35 cross-sectional area of 0.00489 ft2 (see Figure 3-3 for the modelled tower). For the deck supports, only translation in the z-direction and rotations about the z and y axis are allowed. One of the two supports is a roller, as illustrated in Figure 3-3, which allows x-displacement, too. The column cross-sectional areas for the model were increased in proportion to the increased spacing between the columns in the model over the spacing in the original bridge. 3.4 Computer Programs. Two programs have been prepared for this study: Program NEAMAH and Program EIGGRAPH. NEAMAH may be used to evaluate the nonlinear equilibrium response and the n-eigenvalues_ and corresponding eigenvectors of a general space framed structure. The program was an extension of one originally developed at Michigan State University by Jose Lange (21) for the lowest eigenvalue of a curved beam deformable in three dimensional space. It was extended for three dimensional nonlinear equilibrium problems using beam finite elements solution by Jalil Rahimadeh (22). The author's contribution lies mainly in the increased capability and efficiency in handling load inputs, the solution for more than one eigenvalue, and the incorporation of the space truss element in the program. Subroutine BAND was rewritten for program NEAMAH, and for 36 the "EEEEEEEE,-QPdat°d solution," the rotation of the major principal axis was modified. The program coding is listed in Appendix 8. Program EIGGRAPH was developed with the aid of the consultants at the MSU Computer Laboratory to plot the eigenvectors. Plots obtained using this program can be seen in chapter IV. All plots are plotted with scale 1:20 unless otherwise mentioned in the figure. 3.5 Validation. 3.5.1 Introduction. The purpose of this section is to present data for the validation of the computer program (NEAMAH) and 'the procedures which it embodies. Comparisons with published results in the literature are made where possible. A check of the truss linear stiffness with SAPIV is made. The buckling loads of a plane truss-tower, a plane truss and beam tower are compared with the buckling of a cantilever column. The effects of tolerance on the eigenproblem solutions were also studied. 3.5.2 Vertical Stability. For a validation of the program to solve problems involving vertical stability, the FHAB bridge was used. Equilibrium solution was obtained by use of the fupdated-Lagrange" procedure. Plotted in Figure 3-11 as functions of the loading are the quarter point deflection 37 and the determinant of the stiffness matrix. The equilibrium solution may be used to identify the bounds of the buckling load for the structure. Instability may be assumed to occur when the determinant_vanishes. _Itmismnot possible using the available equilibrium solution procedure to actually pinpoint the exact buckling load due to the use of finite _loadkincrements. Instability occurs between two successive load increments in which the first isfistable_and the ”second is not. Using smaller load increments would decrease (the bounds, yet the solution cost would increase very significantly. The bounds are plotted in Figure 3-11 as a dotted line. On the other hand, an eigenvalue solution was obtained for this problem and plotted as a horizontal straight line in the figure. It is seen that the eigenvalue solution lies within the bounds of the equilibrium solution. Since the two solutions were essentially independent, the results lend credibility to both. 3.5.3 Lateral Stability Attempts were made to compare the lateral buckling load using Athe eigenvalue solution with existing data available in the literature. The comparisons included results given by: (1) Equation 16.25 in Reference 10 for out-of—plane buckling of single arches, (ii) Lars Ostlund (13) for the buckling of two ribs braced with beam element, and (iii) Tokarz (4) and Almeida (15) for two braced ribs. 38 3.5.3.1 Equation 16.25 in Reference 19_ In the book: "Guide to Structural Stability of Metal Structures" (GSSMS (10)) edited by B. Johnston, Equation 16.25 was presented for estimating the critical load for the out-of—plane buckling of a single parabolic arch rib loaded uniformly. The effect of the in-plane flexural rigidity is not considered. Three arches were used for comparison. (1) Arch:A as presented in Figure 3-12, (ii) the FHAB, and (iii) the MCSCB (all as single arches). The results are given in Table 3-3. The equilibrium solution of arch-A is also presented in Figure 3-13. The results are seen to compare well. 3.5.3.2 Ostlund's Data Two examples were considered, both for structures with two ribs braced with cross beams only (Vierendeel type). Figure 3-14 gives the arch dimensions, the loading condition, and material properties. The properties of the cross-sections of the various elements are presented in Table 3-4. The comparison of results is listed in Table 3—5 for examples one and two. The tabulated values are for C = qoLz/EIo where qo is the critical axial compression at the crown. The three values of c that appear in Table 3-5 are due to different approximations employed by Ostlund as noted in Table 3-5. Note that three types of "bridges" were considered. The "No-Deck" type represents 'the two braced ribs with 39 loads applied at the panel points of the lribs. The "Through Bridge" type denotes the case in which the loads are applied to the arch ribs through rigid hangers that are jointed to a laterally rigid deck (from which the vertical load is supposed to be applied). The "Deck Bridge" type denotes the case in which the loads are applied to the arch , ribs through rigid columns that are joined to a laterally rigid deck. Analyses of the latter two types are known as "tilted load" effects as discussed previously. It is seen that Ostlund's results are generally not sensitive to the approximations used. Where the approximation resulted in appreciable differences, it is interesting to note that the results obtained using NEAMAH lies in the range given by Ostlund's approximations. ‘It may be noted that a substantial difference exists in values of the first buckling load for the "Through Bridge" between Ostlund's data and the NEAMAH results. The difference may be explained by the fact that the much lower value given by NEAMAH corresponded to a mixed buckling mode in a truly three dimensional solution. Ostlund's solution corresponded to a prescribed purely lateral buckling mode; i.e., it probably missed this lower mode. 3.5.3.3 The Data of Tokarz and Almeida. Many tests were run by Tokarz (4) for a single rib, two ribs braced at the crown with one beam and two ribs braced at a number of panel points. The results of two tests were compared herein. They correspond to test number 4O 27 and number 33 in reference 4. Each test was conducted on two aluminum ribs, 2024-T3 with modulus of elasticity E I 10.7 x 106 psi and Poisson's ratio I 0.3. The ribs were 0.192 in. thick and 1.5 in. deep. The ribs were fixed at the two end supports and real deck was constructed. Test number 27 was braced with 15 equally spaced round bars of 3/32 in. diameter and test number 33 was braced with 15 equally spaced round bars of 5/32 in. diameter. The two structures were also analyzed by Almeida (15). Comparison of the results is listed in Table 3-6. It is seen that the agreement is generally quite good. 3.5.4 Truss Elements. To check the program for the addition of the truss elements SAPIV was used. Two problems were solved. The first was a system built of fourteen truss elements and eight nodes subjected to one concentrated load. Program NEAMAH and SAPIV gave the same results for both linear displacement and axial forces. Secondly, a system consisted of four columns (beam elements) supporting a horizontal truss grid of five elements subjected to one concentrated load at one of the corner nodes. Again the agreement was total. For the study of the truss elements in a buckling problem, the buckling load of a tower built of truss elements (see Figure 3-15) was calculated by the eigenvalue program and compared with- the Euler buckling load of a cantilever column. Then the columns in the tower were 41 replaced by beam elements and the buckling loads computed again. For the truss tower (see Figure 3-15), its buckling load may be estimated as that of a column with a moment of inertia It = 2At(S/2)2 where S is the width of the truss. For beam-truss tower, similarly the buckling load may be estimated as that of a column with a moment of inertia IBT I It + IB where IB is twice the moment of inertia of the beam element. Table 3-7 presents the buckling loads obtained by NEAMAH of the truss tower and the truss-beam tower as described in Figure 3-15 and the equivalent Euler column .buckling loads. The buckled shape of the truss-beam tower is illustrated in Figure 3-16. The buckled shape of the truss tower is similar. 3.5.5 Tolerance Effect. 3.5.5.1 Equilibrium Solutions. Three different tolerances were used for the purpose of checking the tolerance effect on equilibrium solutions. The values of the tolerances are 1x10'2, 1x10'5, and 1x10'7. A single rib of the _MCSCB was used, subjected _to a constant uniform load of 5.858 kips/ft over“ the full span and an incremental load of 0.571 kips/ft on i' one half of the bridge span. The obtained data showed that 5, f..- .4- the nonlinear response was not affected by the tolerance values’used. 42 3.5.5.2 Eigenvalue Solutions. For this study, the buckling load of a pin-ended column was considered. The column is 120 ft. long with xxx = 35.99 ft4, Izz = 3.947 ft4, and KT = 10.97 ft4. It was represented by 12 beam elements. The results indicated that the_ eigenvalues as such were_notwaffected by the value of the tolerance used in the solution. however, the sequencer in which they were “ha—.fiv’ successively generated by the procedure was. Thus the smaller the value of the tolerance,r the better or more reliable the sequence of the eigenvaluesvis. Table 3-8 lists the first five eigenpairs corresponding to two values of tolerances used. Note that for the case of the higher tolerance, the third mode was obtained ahead of the second mode, the fourth mode was not even in the picture. For the lower tolerance, the first three modes were produced in the proper order. The fifth was generated before the fourth. For this simple example structure, the proper values for the modes are known a priori. This is not generally the case for a complex structure for which the eigenvalues are being sought. Therefore, one must use the method with some .33351991 _although for all cases considered herein the first “med Rnwas generated first. It should also be mentioned that irrespective of the order of generation, the mode shape qbtained’ always corresponds to the correct buckling load. Of course, _ the buckling mode itself contains much information. CHAPTER IV BUCKLING LOADS AND MODES Winn. Elastic buckling loads, as a type of limit load, are of interest by themselves. In addition, as was explained in Chapter II, they are needed for the amplification factor method which may be used to estimate the nonlinear response. As discussed earlier in Chapter I, extensive data are available on factors affecting the buckling loads. However, previous researchers had studied either lateral buckling or in-plane buckling, i.e., where each case is treated sepa- rately. Furthermore, all previous works on three dimension- al elastic stability involved cross beam bracing and asso— ciated parameters only. The new aspects of the stability problem of arch bridges that are considered in this research include: (1) the general stability of arch bridge in three dimensional space without having to limit the problem a priori to either in-plane or lateral buckling; (2) the effect of the finite stiffness of the deck on in-plane and out-of—plane stabili- ty; (3) the different types of bracing in the bridge system and their effects on the overall stability. The data obtained here will be given in terms of as = wc/wy, where wc is the uniformly distributed buckling load, 43 44 and wy is the uniformly distributed yield load evaluated as follows: 8f w 3i Y L2 where f is the rise, 0y is the yield stress, A is the average cross-sectional area of the arch rib, and L is the span. The tolerance used in the eigenvalue solution is l x 10“. Ai2__Brasin9_BsLneen_Bihs_:_flrasing_zattsrns. Different types of bracing patterns have been used in practice. In this study, several different common patterns of bracing are compared based on equal volumes for all patterns. The different patterns used for this study are defined in Figure 4-1, which includes (a) x-truss, (b) K-truss, (c) Diagonal, and (d) Transverse bracings. The arch ribs used for this study were modelled from the Cold Spring Canyon Bridge as described in Chapter III. 'The total volume, V, of the bracing members for the x-truss bracing case was derived using the same modelling approach presented in that chapter using eight panels.' This resulted in cross-sectional area of diagonal members, AD 3 0.341 ft.2 and transverse members, AT - 0.341 ft.2 for the eight-panel model, the volume of the x-truss case is: v a 16 AD\[32 + ,2 + 7 AT 8 a 778.46 ft.3 (4.2.1) 45 in which 3 (a 70 ft.) andk.(- 87.5 ft.) are the spacing between the two ribs and the panel length, respectively. For the K-truss bracing case, the volume is: v = 16 AD ‘/(s/2)2 + 22 + 7 AT 3 = 778.46 ft.3 (4.2.2) Using the same value for AT as for the above x-truss case, AD evaluated from equation 4.2 is equal to 0.4055 ft.2 Following a similar procedure, the volume of the diag- onal bracing case is given by: v = 16 I‘m/32 + 22 (4.2.3) for equal volume, AD - 0.4342 ft.2 For transverse bracing, the volume is V s 7ATS and the cross-sectional area of each transverse beam is AT . 1.5887 ft.2 The beam flexural and torsional stiffnesses were calculated assuming that the cross-sectional depth is twice as much as its width and its thickness equal to 1/12 its width. Five buckling loads were obtained for each bracing pattern. The results are presented in Table 4-1. The buckling loads are given in terms of wé/wy in which, as previously, we is the uniform buckling load and wy is the uniform yield load. Some of the buckled shapes are illus- trated in Figures 4-2 through 4-13. The buckled shapes are also summarized in Table 4-1. The following abbreviations were used for that purpose. 46 In-plane: The arch buckled “mainly" in the x-y plane. ('mainly' implies that the maximum displace- ment in that plane is at least one order of magnitude larger than those in the other orthogonal planesd Out-of—plane: The arch mainly buckled in the z-direction. Lat.-tors: The arch buckled in a lateral-torsional mode. 3-D mixed: The buckled shape is three-dimensional and of a nondescript form. Sym.: Symmetric. Anti-sym.: Anti-symmetric. An examination of the data presented indicates the following. 1. The lowest in-plane buckling loads and corresponding modes are essentially the same for all cases. This shows wigggwsst in-plane buck11n9_109§48 independsngof lateral bracing1 The lowest in-plane buckling mode is the first mode (among all modes--in-p1ane and otherwise) in all cases except for the beam-bracing case where it is the fifth. For the beam bracing case, the first four modes are all of the lateral buckling type with buckling loads less than the lowest in-plane buckling load. Thus the beam bracing is the weakest pattern. The second in-plane buckling loads are also largely inde- pendent of lateral bracings. ,For the x-bracing, the second in-plane buckling load corresponds to the second mode, for the K— and D-bracing it corresponds to the 47 third mode. The lowest lateral-torsional mode is the third for the X-bracing (wc/wy x 1.447) and second for the K— and D-bracings (we/wy - 1.271, and 1.218 respec- tively). These data would lead to the observation that whenever lateral displacements_are involved inwthe bugklingmode the beneficial effects of the bracing seem toflincrease in the order of D-truss, K-truss to x-truss. {9.929996% 1. The pattern of lateral bracing is essentially negligible if the buckling mode is in-plane. 2. If the buckling mode involves appreciable lateral dis- placements, then the effects of the bracing would in- crease in the order of B-bracing,‘D-truss, K-truss, and x-truss patterns. The differences between beam and truss bracings are substantially larger than those among the different truss-bracings themselves. I 3 E i E l E'l _ E l E E . . The preceding section - 4.2‘considered the effect of bracing pattern on the buckling behavior of two braced ribs. In that study the amount of bracing (as referenced by the total volume) of bracing material was fixed. In this sec- tion, the effect of the amount of bracing is considered. Two bracing patterns are considered: (a) x-truss bracing for the MCSCB ribs, and (b) Beam-bracing for the FHAB ribs. For each case the amount of bracing will be varied. 48 I 3 J MCSCB Bil l 'I] “_I I . )- For the case of truss-bracing, a study of the available data on several existing bridges (see reference 20, chapter 13), showed that the ratio’of the bracing cross-sectional area, Abo' to the rib cross-sectional area, AR, is approxi- mately 1/6. Using Abo as a reference cross-sectional area, the following ratios were employed: Ab/Abo :- 0.001, 0.01, 0.1, 0.25, 0.5, 1.0, and 2.0. Since, as indicated by the preceding section, lateral bracing has little effect on in- plane buckling, and to save computing time, the in-plane moment of inertia of the ribs were increased by 20 times in order to force the out-of—plane buckling to be the lowest mode of buckling. The results of the MCSCB x-truss bracing are plotted in Figure 4-14 and the buckled shapes are presented in Figures 4-15 through 4-20. Observation of the presented data leads to the following conclusions: 1. Figure 4-14 shows that practical variation of the brac- ings stiffness does not affect significantly the out-of- plane buckling load. , 2. As the (Ab/Aha) ratio is reduced the buckling load of the bridge approach the case of a single rib. Such reduction is very significant when the ratio is less than 0.1 and the buckled shape changes from anti-symmetric out-of- plane to symmetric out-of—plane. 3. In the range of Ab/Abo = 0.1 to 1.0 the buckling load increases linearly with bracing. In the parameter range 49 of 1.0 to 2.0 the buckling load increases at a higher rate. Comparing the buckling load of an X-truss braced bridge in Table 4-1, the out-of—plane anti-symmetric mode were (wc/wy s 1.45) and that of Figure 4-14 were (we/wy - 1.864), one notices that the latter is higher even for less amount of bracing. This is merely due to the in- crease in the in-plane stiffness of the bridge. Such an effect is not considered in the formula given by refer- ence (10), as discussed in Chapter III. Within practical range of bracing, for a bridge braced with x-truss bracing the lowest out-of—plane buckling' mode is anti-symetric, whereas, for beam-braced bridge the lowest out-of-plane buckling mode is symmetric. (see Ostlund, Tokarz, and the FHAB). This may be due to: (1) For the symmetric mode theiouter rib would be subjected to tension and the inner one to compression, while the anti-symmetric mode could take place inextensibly, and (2) for the case of diagonal bracing, it would appear that the diagonals would be strained to a greater degree in the symmetric mode than the anti-symmetric mode (See Figure 4-21). Waging. In this section, the amount of bracing and spacing effect on the bridge buckling is studied. The bridge is not forced to buckle out-of—plane. As presented earlier in Chapter III the lowest buckling load is in-plane. The data for various amounts of bracing is compared with the single 50 rib where no bracing is used. Results are presented in Table 4-2. Observation of the data leads to the following: 1. The data presented in Table 4-1 and 4-2 shows that the lowest buckling load for a properly designed bridge is in-plane. 2. The table showed that small variation in the.amount of bracing did not affect the in-plane buckling until the bracing was reduced to 1.258. Then the bridge buckled out-of-plane. 3. Increasing the spacing by three times did not affect the in-plane buckling. 4. With 1.25% reduction in the bracing the out-of-plane buckling load was still higher than that of a single rib. AaA__Inznlan£_§££££L_Q£_D£Ck. As was mentioned in Chapter II, a deck situated above the arch rib reduces the out-of-plane buckling load. No data are available on the tilted load effect on in-plane stability, and no actual study has been reported on the finite deck stiffness effect. The latter factors will be studied in this section. To conduct the study, the MCSCB, 8-panels, single rib, with and without a deck was used. Five cases of two braced ribs were examined: 1. A reference case of no deck. 2. A truss deck, load applied on the deck-~0nly tilted load effect is expected, since the truss deck has no in-plane (with reference to arch ribs) stiffness. 51 3. Beam.deck, load applied on the deck--Both tilted load and deck stiffness effects are expected. 4. Beam deck, load applied on the rib--Only the deck stiff- ness effect is expected. 5. Truss deck, load applied on rib--No deck effect is ex- pected. Results are presented in Table 4-2 in terms of a. An examination of the data leads to the following observations: kW 1. From cases one and two, the tilted load effect.isiesti- mated as T1 = 0.6714 - 0.5184 - 0.1530 2. From cases three and four, T2 = 0.7297 - 0.5612 a 0.1685 Thus, the tilted load effects is for T1, 0.1530/0.6174 = 24.8% and, for T2, 0.1685/0.7297 = 23.1% Bl__Dssk_£tiffnsss_£ffsst 1. From cases two and three the effect of deck stiffness is estimated as $1 a 0.5612 - 0.5184 a 0.0428 2. From cases four and one, deck stiffness effect may be estimated as $2 = 0.7297 -'0.6714 = 0.0583 52 Thus, the deck stiffness effect is, for $1, 0.0428/0.5184 a 8.3% and, for $2, 0.0583/0.6714 - 8.7%. It may be interesting to note that the ratio of the deck to the rib flexural rigidity is 0.10. It had been mentioned (19) that the buckling load should be increased in direct propor- tion to the sum of the bending rigidities of the ribs and the deck. The preceding data indicated that the increase is somewhat less than that. D W Buckling In the preceding section, the deck effect on the in- plane buckling load of the arch bridge was studied. All columns were assumed pinned. Therefore, no shear transfer is assumed between the deck and the arch ribs in the longi- tudinal direction. In practice, arch bridges are designed so that such shear transfer would take place. This, of course, depends on the connection between the ribs and the deck. Such connection is studied in this section. Shear connections usually used in practice are: 1. Rigid 'moment' connection at one or more intersection panel point (see Fig. 4-22a). 2. In-plane spandrel bracing at one or more panels (see Fig. 4-22b), and 3. The ribs and the deck are rigidly connected at the crown, i.e., for this case the deck is at the same elevation as the crown is. The MCSCB, 8-panels, with deck as presented in Chapter III was used. 53 The results obtained for rigidly connected columns and spandrel bracing are shown in Table 4-4 and also plotted in Figure 4-23. Some of the buckled shapes are presented in Figures 4-24 through 4-28. In Table 4-4 case one denotes the reference case, which is the case of no shear transfer. Cases two and three represent perfectly rigid connection of the crown column and all columns respectively; For this study the moment stiffness of each rigid column was varied. Studying the present results indicate the following: 1. The buckling load almost doubled with only the crown column rigidly connected. 2. An increase of the bending stiffness by one order of magnitude would force the lowest buckling mode to the symmetric mode. . 3. As expected, by rigidly joining all ends of all columns, the lowest buckling load increased by 40% over the case of only the crown column is rigidly connected. 4. Cases four and five refer to spandrel bracing of one and two middle panels. The case AD/Ao s .325 corresponds to the use of a 1 5/8 in. rope. It is obvious that the use of spandrel bracing increases the buckling load (PC) but there is no substantial difference between the different. cross-sectional area or when the spandrel bracing is extended to the case of two braced middle panels. The effect of column heights is shown in Table 4-5. The column heights are defined by the crown column with all other columns varying to conform to a horizontal deck. All columns.are pinned unless otherwise noted. 54 It is seen that as the columns shorten Pc decreases. This is due to the P-A effect. The shorter the column, the greater the de-stabilizing effect Or the P-A effect. When the ribs and deck are joined together with no column at the crown the Pc increases because the bridge~is forced into a symmetric mode, likewise for the case the h/ho,- 0.08 but the crown column is rigidly connected where essentially the same Pc is achieved with the same symmetric buckling mode. W W. The two towers in a deck bridge support the deck which is situated above the ribs and meet with the two ribs at the foundation. The two towers are part of the structural system and their rigidity should be significant to the- overall stability of the bridge. For vertical buckling, the towers are very rigid and no significant effect is expected. For lateral buckling the two towers work as laterally loaded frames. This makes lateral stiffness of the bridge affected by the lateral stiffness.of the tower. Theideck in-plane effect was discussed in section 4-4. The out-of-plane ef- fect of the deck has been extensively studied in the litera- ture, yet all works looked at the deck as very rigid and correspondingly only tilted load effects wereiconsidered. The tilted load (P-A) effect is very significant but the deck stiffness would seem to be a very important part of the deck effect. The latter effect can only be studied by assuming that the deck is not rigid, and that it actually' undergoes elastic deformations in three dimension. 55 Transverse bracing (see Figure 3-3) is often used for steel bridges. Transverse bracing adds to the global stiff- ness of the bridge. The bridge used for this study is the MCSCB, 4 panels (see chapter III - Table 3.1). The deck and the tower was modelled and included in the same table. The transverse bracing was studied once by bracing the MCSCB at the crown intersection panel only and by uniformly bracing the bridge (i.eu, bracing every intersection panel except at sup- ports.). To study the deck lateral stiffness the flexural rigid- ity of the deck, Iyy (as modelled in Chapter III), was reduced by 10, and increased by 10. Similarly the lateral stiffness of the“ tower was reduced by 10, and increased by 10. The results are available in Table 4-6 and buckling is presented as a. The buckled shapes are presented in Figure 4-31 through Figure 4-40 and the deck lateral stiffness and the tower lateral stiffness as compared to the classical rigid deck effect are illustrated in Figure 4-41. A study of the data leads to the following conclusions: 1. The first row in Table 4-6 shows the basic reference case, or the case with no transverse bracing, and hence no shear transfer. The lowest buckling load (PC) is a - 0.169 and both the deck and the rib buckled of the same magnitude, the second Pc is a a.- 0.613 which is in-plane anti-symmetric including deck effect (P—A and deck stiffness). The 56 third Pc is w a 0.899 out-of-plane anti-symmetric with no in-plane buckling and the tower is involved in some lateral deformation with the deck. The fourth PC is a = 1.155 and is just like the third except with additional in-plane buckling which makes it generally three- dimensional. The fifth mode has a similar configuration. The rows 2a and 2b correspond to transverse bracing at the crown intersection panel only and at all intersection panels, respectively. The previous half wave symmetric mode out-of—plane (a; - 0.169) was eliminated by the bracing and the lowest PC (w - 0.613) corresponds to an in-plane mode. The lowest out-of—plane is anti-symmetric with fi's 0.9. The effect of the deck lateral stiffness is shown by rows 3a, 3b, and 3c, which correspond to lateral stiffness of 1/10, 10, and perfectly rigid (using the classical solu- tion known as the tilted load effect). All cases are without transverse bracing. In conjunction with case one, the deck lateral. stiffness increase the lateral Pc but even with infinite- ly rigid deck the lateral PC is smaller than that of in- plane buckling. The effect of tower lateral stiffness is shown in row 4a, and 4b. As expected, the tower lateral stiffness in- creases Pc' However, the relation is not linear. The results are presented in Figure 4-41 which shows that increasing the tower lateral stiffness by 10 times, Pc is increased by only 12%.' 57 In summary: Transverse bracings are very effective and it seems important to have at least one intersection-panel braced. Deck and tower stiffness do matter. But the influence is not linear. Existing designs seem to be effective and large increase of their lateral stiffness would not greatly increase the lateral stability of the existing designs. CHAPTER V NONLINEAR RESPONSES 5al__1nilndnstlnn. The amplification factor method as a means of estimat- ing the nonlinear response to loads in the lateral, longitu- dinal, or vertical direction is studied in this chapter. Results are compared with nonlinear equilibrium solutions as obtained by use of program NEAMAH. Wanna. 5a2a1__N9nlin§3L_£Qnilihrinm_591ntion. Using the FHAB bridge (Chapter III), a solution was obtained for a constant uniformly distributed vertical load, wf, and variable uniformly distributed lateral load, wa, (see Figure 2-3c). Results are presented in Figure 5-1 and 5-2. Figure 5-1 shows that at the crown, for the fixed wf and increasing lateral load wa, the vertical displacement increased nonlinearly, but the lateral displacement was linear. Figure 5-2 shows that the latter type of linearity held not only at the crown but also at other points along the bridge. It should be noted that the above-mentioned the linearity is with reapect to w For a given w a' a' 58 59 response with respect to varying wf would obviously be nonlinear. W. The accuracy of the amplification factor method may be considered using the same combinations of lateral and verti- cal loading conditions as in the preceding section. Three different cases were employed, each for a different value of fixed wf/we and variable w The amplified responses were a' compared with 'actual' nonlinear response (obtained from equilibrium solutions). The AF was computed from the following: 1 AP =-——————— (5.1) l - a where a a wf/we, wf is the fixed vertical uniform load, and we is the I'compatible buckling load," (see Chapter II). For the FHAB used for this analysis the lateral buckling load was we/wy - 2.264. This is not the lowest Fe for the bridge model. It is the second. Knowing the values of wf and we, the lateral nonlinear response Rn' due to the additional load we, was computed from: in which RL is the linear response. A reference lateral load due to a wind pressure at 100 mph wind velocity was used for this presentation. Results are presented in Table 5—1. The presented data shows the 60 ‘values of wf/we, the Amplification.Factor, the linear re- sponse, the nonlinear response, and the ratio of the esti- mated to the ”actual" nonlinear response. The data show that the amplification factor method yielded good estimates of the nonlinear responses in the lateral direction. Was. 5l3ll__Nsn1insar_§gnilibrium_fiulutign. In a similar manner to that presented in the preceding section, the nonlinear equilibrium solutions for longitudi- nal displacements were obtained. The FHAB bridge was used with several cases of fixed load, wf, each accompanied by variable longitudinal additional load, we (see Figure 2-3b). Results are presented in Figure 5-3 for the crown point. It is seen that for a given wf, and increasing wa, the vertical displacement increased nonlinearly, but the longitudinal displacement was linear. This situation is similar to that of the lateral loading as discussed in the preceding section. WWW. Using the FHAB bridge model and the procedure outlined in section 5.2.2 for estimating the nonlinear response, the amplification factor method for nonlinear longitudinal re- sponse was considered. In this case the compatible buckling ' mode is the lowest in-plane anti-symmetric mode with a corresponding buckling load equal to wf/we a 1.052. .This is the lowest critical buckling load for the FHAB bridge model. 61 The results are presented in Table 5-2 where the same reference wind pressure load was used as previously. The data shows that the amplification factor method again provided good estimates of the maximum longitudinal responses for the arch bridge model. MW. 5a1a1__fisnfizal. In the preceding sections, wf, was applied in the vertical direction but the responses considered were ortho- gonal to the vertical direction, i.e., the linear and non- linear responses were in the same direction of "a“ Since the response considered in this section is vertical, the effect of wf is to be included in the computation of the linear response, ine., Depending on the loading pattern, the compatible buckl- ing load would be different. In Chapter IV it was found that the deck-rib connection condition significantly af- fected the buckling mode and load. USing the MCSCB bridge, eight panels, with deck, three cases were considered for studying the amplification factor method as applied to vert- ical loading. They are discussed in the following. W. In this case the lowest buckling mode was anti-symmet- ric, we/wy =- 0.561, on values used were 0.15, 0.25, and 0.5 62 and the loading condition was presented in Figure 5-4a. The results are presented in Table 5-3 for the quarter point under the additonal load wa. It is seen that the AF method provided good estimates with an.'error' generally less than 10%. The AF method tended to underestimate the maximum response as the value of wa increased. For this type of connection, two subcases were consi- dered depending on the loading pattern. W. The loading condition for this case is shown in Figure 5-4b. The corresponding buckling load was we/wy - 1.065. The values of on used were also 0.15, 0.25, and 0.5. The results are presented in Table 5-4 for the quarter point under the additional load w It is seen that the 3. accuracy of the estimates provided by the AF method was somewhat lower than the previous case. But, they may still be considered good if the loads wa/we are not greater than, Bay, 0.10. 5aAl3a2__Nanni£QLm_LQadinQ. The loading condition as shown in Figure 5-4c, with the ratio of the left half-span loading to the right half-span loading equal to 1.2, and the applied load parameter w is incremented. Note that w is the only applied load (compar- able to w + wf), and a = w/we. This means that the a 63 “amplification factor," AF, is variable for each load incre- ment. The range of a used is 0.15 to 0.55. The results are shown in Table 5-5 and Figure 5-5 also for the quarter point under the larger loading. The pre- sented data show that at least for this pattern of nonuni- form loading the amplification factor method produces good conservative approximations to the nonlinear responses. The range of validity in this case is larger than the previous C3868 . 514aL_Jnhs_anthxwuflusidLLJkummsied. The loading pattern for this case is presented in Figure 5-4b. The compatible buckling mode is symmetric (also corresponds to the lowest buckling load, we/wy - 1.468). The values of a used were 0.147, 0.254, and 0.423. The results are presented in Table 5-6, for the crown, which shows that the amplification factor method gives good estimations for the nonlinear responses. For a 8 0.147, the error was about 8% to 10%; for a a 0.254, it was between 2% to 9%, and for a - 0.423 it was between 4% and 16%. CHAPTER VI SUMMARY AND CONCLUSION Mm. W. The objective of this research was to examine the problem of elastic stability of arch bridges and to consider a simple approximate method for estimating the maximum non- linear response. The method, which uses the linear response and an amplification factor (function of the buckling loads) is called the “amplification factor methodJ' Computer modelling was employed to compute the buckling loads and modes and to obtain the nonlinear equilibrium solutions needed for checking the above mentioned amplifica- tion factor method. The bridge models used were three dimensional and quite complete, each including two ribs, bracings between ribs, deck system, longitudinal and lateral bracings between the deck and the ribs, and end towers. A computer program, NEAMAH, wasideveloped for use in this study through modification and expansion of certain available ones. The program was validated by extensive corroborations with known data. The obtained results were in two groups, summarized as follows. 64 65 5all2__Bucklins_hnada_and_fl9des. (1) Effect of rib bracing--It was found that truss bracing, as compared to beam bracing (Vierendeel), could increase the lateral buckling load by as much as three times for the same amount of bracing material. (ii) Longitudinal shear transfer--Providing a rigidly connected column to transfer shear between the deck and the ribs, rigidly connecting the deck and the ribs, or use of spandrel bracing increased the in-plane buckling load two to three fold. Such shear transfer should be provided for the stability of the bridge in the vertical plane. (iii) Effects of the deck--A deck situated above the ribs has a positive and a negative effect on in-plane buckl- ing. It provides an additional stiffness in the vertical plane. This added stiffness increased the buckling load. But the increase was a little less than what the ratio of the deck to the ribs flexural rigidities would indicate, as was suggested by some investigators. The “negative“ effect of the deck is analogous to the "tilted load' effect for lateral stability (similar to the so-called 'P—A effect"). For the MCSCB model considered, such effect was about 23 to 25% of the overall rib buckling load. The stiffening effect was 8-9%. Thus, the softening effect due to the deck was much greater than the stiffening effect. (iv) Lateral stiffness of the tower and deck-~Corre- sponding to a reduction and an increase of the deck lateral stiffness by ten-fold, the buckling load was reduced by 80% and increased by 55%, respectively. Similarly for the 66 tower, the lateral stiffness was reduced and increased by ten-fold and the buckling load was reduced by 50% and in- creased by 12% respectively. (v) Effect of transverse bracing--Transverse bracing between the ribs and the deck increased the lateral buckling strength markedly. For even with one transversely braced panel, the buckling load would be increased by more than three-fold over the case with no transverse bracing. W. The lateral nonlinear response was presented for wf/we 0.0627-0.2547 and wa/wloo 3 0.5-83. For the wide range of data presented for the FHAB (beam bracing only) the 'error' was no more than 2%. The longitudinal nonlinear response, obtained for wf/wc =- 0.’0489-0.4888 and wa/w :- 3.3-52.3, had errors that were 100 no more than 2% in each case. The error can be 20% for wa/wloo higher than 52.3. The vertical response was studied for three cases of deck-ribs connections. (i) All Columns Pinned--For wf/we :- 0.15-0.5 and wa/we -:0.0325-0.1625, the error in estimating the vertical.non- linear response was not more than 10%. (ii) Crown Column Rigidly Connected (All Other Columns Pinned)--Two subcases were considered. (a) For a uniform wf/we = 0.15-0.5 and wa/we a 0.0325 - 0.1625, the data were not as good as previously. The error approached, in some cases, 20%. However, for 67 ‘wa/we less than 0.10, it was less than 10%. For the case where wf/we a 0.5 and wa/we a 0.1625, the error jumped to 38%. (b) In this case, the buckling load was based on the same nonuniform load pattern--As sampled for a range of w/wc a 0.1509-0.5460 all the amplified responses were conser- vative and the maximum error was about 11%. (iii) Deck and Ribs Rigidly Connected--The data were obtained for wf/we - 0.147-0.423 and wa/we .. 0.0325-0.1625. the error was not more than 10% except that for the case wf/we a 0.423 and wa/wC - 0.0325, the error increased to 16%. In the application of the amplification factor method, it is essential that the amplification factor be computed using the ”compatible buckling load' (the buckling load that corresponds to a buckling mode conformable to the response under consideration). Wants. Because of cost, only two bridge models were used in this study, but the qualitative aspects of the results should be applicable to deck arch bridges in general. The buckling loads and modes indicated that the problem should be considered as one of a three-dimensional system so as not to miss any mixed mode buckling load that cannot be pre- dicted by a formulation that rules out mixed mode a priori. Current design practice seems to provide some form of shear transfer between the deck and ribs, for example, using 68 bracing members or rigid connections. The practice seems adequate so far as elastic buckling is concerned. The stiffness of the end towers also seems adequate. The use of the amplification factor method for the estimation of the nonlinear response appears to be quite promising for practically all types of loading. Of course, more data are needed to extend and/or establish the range of validity. This study has focused on geometric nonlinearity of the response of deck type arch bridges. The critical buckling loads have been presented in terms of the yield load. It was obvious that.in some cases yield would occur before elastic buckling could take place. Therefore, it should be - natural that material nonlinearity be considered for future research. 69 Table 3-1 Cross-sectional PrOperties of MCSCB Four Panels, with Deck (for element number identification refer to Figure 3-4 a, b, and c.) Element number A Ixx Izz KT 1 2.6559 35.99 3.9390 '21.o100 2 2.9058 41.58 4.1260 24.7500 3 0.7860 3.72 2.1800 1.3600 4 0.9000 -- —— -- 5 1.0891 1.27 0.4165 0.9661 6 1.0891 4 —- -- —— 7 14.7800 -- -- -- 8 14.7800 -— , -- -- 9 0.00489 -- -- -- Table 3-2 Cross-sectional Properties of MCSCB - Eight Panels (for element number identification, refer to Figure 3-4 d). Element number A Ixx Izz KT 1 2.4000 30.28 3.7400 24.88 2 2.9058 41.58 4.1260 25.00 3 3.0300 44.52 4.2200 25.30 4 2.6559 35.99 3.9390 24.95 5 0.7860 3.72 2.1800 1.36 6 0.3264 -- -- -- 70 Table 3-3 Values of Critical Loads for Lateral Buckling (Kips/ft.) Ratio By By of equilibrium eigen eigen solution value GSSMS value to armature hounds— snlutinn 1511.. 1.6.2.51 SEEMS.— ARCH-A 8.00 a 9.116 7.8670 8.2600 0.92 FHAB -- 4.5576 4.8738 0.94 MCSCB -- 1.8381 1.9929 0.92 Table 3-4 Cross-sectional Properties for Ostlund's Arches (see Figure 3-14). Elements A xxx 122 KT 1, 2, 7, 8 111.803 1,164.6172 . 931.6946 1,716.9380 3, 4, 5, 6 180.278 4,882.5528 1,502.3167 3,925.8346 9, 10, 11 60.500 152.5104 610.0417 418.8794 1, 2, 7, 8 50.990 441.9100 106.2300 294.3000 3, 4, 5, 6 58.309 660.8240 121.4770 355.0300 9, 10, 11 12.000 4.0000 36.0000 13.9861 Transverse beam bracing minor principal axes orientation coincides with the average angle of the two adjacent rib angles of inclination. 71 .cooumo 12on mange on» no 220323 H632... on» 5332; ed .93 3. can .3 mm 02mm .3 .cocummwumfic mesons one no ocwfim Hmofiuuo> onu ca auficfimfiu amusxoam one Any .cmuomammc mueoewam Hmsoa>wocfi mo uommmo acesaoonemome Am. ma.mm hm.wm 1| mo.am hm.oqa m.haa mm.mm em.mm nmEnoz 54 mp baa mm on 11 mm II or 11 mm mm mm An. me on HNH mm “my penaumo .Acofiumuuomoc o>onm mom manna omeosn Home lefldaunw can oflu can Iuoeemm luoeeam once luoeemm ode . It tunes I: uwucm :1 posse lance unseemm mm.mm mm.m¢ ma.mw >.¢¢ ~w.>m swamoz can can can can nuoeemm Iuoeemm Iuoeemm can Iuoeemm can Iwucm afiucn lasso luweexm laser luweewm am we cad em mm gov II mm 11 no do «a we hm An. mm be mad mm on Am. . cesaumo n unmaunm 0005 docs woos oboe oboe woos woos woos can uma can umH cam uma cam and oono an czouo um mmcfiun goon oz some“: confidau evacuna museum some 1035436 .. u me $32. a; “00925.52... 33 6.2538 a»? 8389.8 mum 633.. 72 Table 3-6 Comparison with Data of Tokafiz and Almeida. Tabulated are the values of weL /EI where we is the critical distributed load per rib Test First mode Second mode No. Source (symmetric) (anti-symmetric) Tokarz 53.4 -- 27 Almeida 46.7 97.70 Neamah 46.92 99.45 Tokarz 79.7 -- 33 Almeida 63.7 114.3 Neamah 77.33 129.99 Table 3-7 Results for Cantilever Column and Tower (All values are in Kips) First Mode Second Mode Truss Neamah 0.448 x 106 3.440 x 106 Tower 6 6 Euler 0.457 x 10 4.113 x 10 Truss-Beam Neamah 0.905 x 106 7.510 x 106 Towe I Euler 0.914 x 106 8.226 x lo6 73 Table 3-8 Tolerance Effect on Eigenvalue Solutions Sequence of Eigensolutions as Obtained Tolerance l 2 3 4 5 1a 3 2 5 7 1 x 10"6 484.46 4,364.61 1,938.21 12,202.34 24,435.82 1 2 3 5 4 1 x 10"18 484.46 1,938.21 4,364.61 12,202.33 7,776.04 a - These numbers are the actual number of the mode. 74 ma.e .mas ~H.4 .mae Hale .mas eHue .maa ale .maa .Ehmlaucw .Emmlwucm .Emm .Eamnuucm .Emm ceramic“ occamlmoluso unnamIMOIuso canHQIHOIuso occamlmoluso 3 «NJ. 3 a 3 0305239 ens .maa sue .mau e14 .mam mu. .maa coxae mcaaxosn .Ehm .Emm .Ehmlfiucm mcmamlmoluso HmooH oceanic“ .muou|.uma ceramics as... amid a 311—. ad musuuno .EMm venue .eam .Emm .Ewmnfiucm ocmamuuonuso olm oceanic“ .nuouukuma mandala“ Manda unuqa anqa asuqa «Nada mussels . ens .mae mus .mae «(a .mae .Ewm nouns .Ewm .Emm .Exmlducs .muounkunfl Dim .muounzumH oceanic“ oceanic“ a a N34 .934 as. mmsuuux woo: woo: woo: woo: woo: cumuumm names euusom cases vacuum amuse msaueum a .Amomcnm pouxosm can 3\oz mo mosam>v acuouumm wagonum ham accumuufio Lou mono; mcwaxosm Hue wanna 75 Table 4-2 Effect of Bracing Cross-Sectional Properties on Buckling Load, FHAB Bridge. Cross-sectional Properties we/wy Buckled A Izz Iyy KT Spac1ng Shape 0.8316 7.997 1.428 1.976 28 1.053 In-plane anti-sym 0.6316 3.441 1.428 1.976 28 1.053 In-plane anti-sym 0.4000 2.000 0.750 1.000 28 1.053 In-plane anti-sym 0.8316 7.997 1.428 1.976 84 1.053 In-plane anti-sym 0.0100 0.010 0.010 0.010 28 0.184 Out-of- plane sym. Single arch rib (no rib-bracing) 0.153 Out-of- plane sym. 76 Table 4-3 Tilted Load and Deck Stiffness Effect All values are presented in terms of a Load and - Case Deck Conditions w 1 0.5714 No Deck 2 0.5184 3 0.5612 57’ 7’ ' ' ‘U 4 0.7297 Beam geck 5 0.6714 77 Table 4-4 Effects of Column Connections on In-Plane Buckling Load. Column ~Column Stiffness connection A/Ao, I/IO Node 1 Node 2 Case type and AD/AO 5 Shape fi Shape Figure 1 All columns , pinned 1.0 0.561 ANT 1.480 SYN 4-24 2 a Columns at 1.0 1.065 ANT 1.480 SYN 4-25 b crown section 10.0 1.480 SYN 1.634 ANT 4-26 0 are rigidly 100.0 1.480 SYN 1.855 ANT d connected 1,000.0 1.480 SYN 1.895 ANT' others are pinned 3 a All columns 1.0 1.423 ANT 1.549 SYN 4-27 b are rigidly 10.0 1.887 SYN 2.435 ANT c connected 100.0 2.422 SYN 2.889 ANT d 1,000.0 2.741 SYN 3.087 ANT 4 a Spandrel 0.325 1.325 ND 1.623 ND 4-28 b bracing for one 1.000 1.426 ND 2.025 ND middle panel 5 a Spandrel 0.325 1.560 SYN 1.805 ANT b bracing for two 1.000 1.656 SYN 2.214 ANT middle panels A0: Cross-sectional area of column (as used for the NCSCB). A: Cross-sectioal area of column (as used for the present study). AD: Cross-sectional area of spandrel bracing. Io: Noment of inertia for column (as used for the NCSCB). I: Noment of inertia for column (as used for the present study). ANT: Anti-symmetric mode. SYN: Symmetric. ND: Nondescript. 78 .o«uuoeemm ”sum .mpoe cuuumeemmluuc< ”92¢ .mpsum acceded one new poms unmade :Esaoo u .momo: now com: me unwed: CEsHoo Hecamfiuo ”on ante 92¢ mcm.~ sum mme.~ 9.: m 924 mee.~ sum aee.s Aeamauv ee.e a amue sum aee.a 92¢ sa~.e .eueead. ee.e s N sum mee.~ aze mom.o Accesses m.e n «mus sum eee.~ aze Hem.e Assesses e.” e a madman cmmsm B madam m macauwecoo o:\s emco N see: H see: was essaoo .psoq meanxosn ocsamch co munmfiem cesaoo mo uooumm mle canes Table 4-6 Tower, Deck, and Transverse Racing Effect on the ridge Stability. 79 Values of 9 s welw ,0 Description First Second Third Fourth Fifth Case of case study node soda node sods node 1 Basic Nodal 9.192. 9.992. 1.199. 1.391 'No Transverse out-of-plane in-plane out-of-plane 3-D 3-D Bracing' syn. anti-syn. anti-eye. anti-syn. anti-eye. 2a Transverst 9.911 .9.929 1.199 Bracing at in-plane out-of-plane 3-D Crown Inter- anti-eye. anti-eye. anti-syn. section Panel Fig. 4-35 Fig. 3-36 2b Tran-vora- 9.9.13 9.993 1.252. 1.539 Bracing at in-plane cutocf-plane cut-of-plane cut-of-plane out-of-plane Every Inter- anti-syn. anti-syn. anti-syn. syn. syn. section Panel Fig. 4-37 Fig. 4-38 3a Deck Lateral M39, Stiffness out-of-plane Reduced by ' syn. 10. Fig. 4-39 3b Deck Lateral 1.291 Stiffness cut-of-plane Increased by syn. 10. 30 Classical 9.3.51 Rigid Deck cut-of-plane syn. 4a Tower Lateral 9.9.5.9. Stiffness axial defores- Reduced by 10. tion in the ' modelled tower Fig. 4-40 40 Tower Lateral 9.1.5.9. Stiffness in— out-of-plane creased by 10. syn. Fig. 4-41 80 Table 5-1 "Lateral" Nonlinear Responses by Equilibrium and Amplification Factor Method. Rn 0.0627 1.0669 0.517 0.1399 0.1488 1.003 2.580 0.6995 0.7444 1.003 20.672 5.5960 6.0085 0.994 113.680 30.7856 32.2625 1.018 0.1274 1.1459 20.668 5.5960 6.4492 0.994 41.344 11.1920 12.8798 0.996 124.000 33.5760 38.2913 1.005 0.2547 1.3417 20.668 5.5960 7.5632 0.993 41.344 11.1920 15.1227 0.993 82.668 22.3840 30.3560 0.989 81 Table 5-2 "Longitudina1"‘Nonlinear Responses by Equilibrium and Amplification Factor Method Wf/Wc AF Wa/Wloo RL ft. Rn ft. RL ' AF Rn 0.0489 1.0514 6.604 0.2100 0.2204 1.051 13.208 0.4188 0.4402 1.000 0.0978 1.1084 3.302 0.1067 0.1174 1.008 6.604 0.2100 0.2327 1.000 13.208 0.4188 0.4639 1.001 0.1955 1.2430 3.302 0.1067 0.1330 0.997 6.604 0.2100 0.2616 0.998 0.2933 1.4150 6.604 0.2100 0.2981 0.997 13.208 0.4188 0.5905 1.004 0.4888 1.9562 3.302 0.1067 0.2104 0.992 13.208 0.4188 0.8135 1.007 52.316 1.6588 3.1955 1.016 105.663 3.3504 5.8073 1.129 132.078 4.1879 6.7760 1.209 82 Table 5-3 “Vertical” Nonlinear Responses by Equilibrium and by Amplification Factor Nethed" (“All Columns Pinned“) wf/we AF wa/wC RL ft. Rn ft. RL ' AF Rn 0.15 1.1765 0.0325 0.3977 0.4620 1.013 0.0650 0.7111 0.8575 0.976 0.0975 1.0246 1.2717 0.948 0.1300 1.3381 1.7058 0.923 0.1625 1.6515 2.1609 0.899 0.25 1.3333 0.0325 0.4538 0.5707 1.060 0.0650 0.7673 1.0233 1.000 0.0975 1.0808 1.4996 0.961 0.1300 1.3942 2.0014 0.929 0.1625 1.7077 2.5302 0.900 0.50 2.00 0.0325 0.5942 0.9442 1.259 0.0650 0.9077 1.6532 1.098 0.0975 1.2211 2.4123 1.012 0.1300 1.5346 3.2248 0.952 0.1625 1.8480 4.0945 0.903 * For quarter point deflection under w 83 Table 5-4 'Vertical' Nonlinear Responses by Equilibrium and by Amplification Factor Nethod* (Crown Column Rigidly Connected; All Other Columns Pinned, wf Uniform) wf/we AF wa/we RL ft. Rn ft. RL ' AF Rn 0.15 1.1765 0.0325 0.5086 0.5845 1.024 0.0650 0.8574 1.0421 0.968 0.0975 1.2062 1.5385 0.922 .0.l300 1.5550 2.0800 0.880 0.1625 1.9037 2.6753 0.837 0.25 1.3333 0.0325 0.6152 0.7511 1.092 0.0650 0.9640 1.2804 1.004 0.0975 1.3128 1.8653 0.938 0.1300 1.6616 2.5178 0.880 0.1625 2.0104 3.2548 0.824 0.5 2.00 0.0325 0.8816 1.2947 1.362 0.6500 1.2304 2.1992 1.119 0.0975 1.5791 3.3300 0.948 0.1300 1.9279 4.8523 0.794 0.1625 2.2767 7.3116 0.623 * For quarter point deflection under w a. 84 Table 5-5 'Vertical' Nonlinear Responses by Equilibrium and by Amplification Factor Nethod* (Crown Column Rigidly Connected: All Other Columns Pinned, Non- uniform Loading) w/wC AF RL Rn RL ' AF Rn 0.1509 1.1778 0.4291 0.4799 1.053 0.1778 1.2162 0.5056 0.5789 1.062 0.2047 1.2574 0.5821 0.6831 1.071 0.2316 1.3014 0.6586 0.7932 1.081 0.2598 1.3510 0.7388 0.9206 1.084 0.2867 1.4019 0.8153 1.0454 1.093 0.3136 1.4569 0.8918 1.1790 1.102 0.3405 1.5163 0.9683 1.3227 1.110 0.4384 1.7806 1.2467 1.9970 1.112 0.4653 1.8702 1.3232 2.2205 1.114 0.4922 1.9693 1.3997 2.4730 1.115 0.5191 2.0794 1.4762 2.7627 1.111 0.5460 2.2026 1.5527 3.1019 1.103 * For quarter point deflection under w a. 85 Table 5-6 I'Vertical'l Nonlinear Responses by Equilibrium and by Amplification Factor Method* (Ribs and Deck Rigidly Connected) wf/wc AF wa/wc RL ft. Rn ft. RL ' AF Rn 0.147 1.1723 0.0325 0.4958 0.6316 0.920 0.0650 0.7705 0.9694 0.932 0.0975 1.0453 1.3239 0.926 0.1300 1.3199 1.6966 0.912 0.1625 1.5947 2.0889 0.895 0.254 1.340 0.0325 0.6485 0.8884 0.979 0.0650 0.9241 1.2738 0.973 0.0975 1.1998 1.6818 0.956 0.1300 1.4754 2.1147 0.935 0.1625 1.7511 2.5750 0.912 0.423 1.733 0.0325 0.8971 1.3352 1.164 0.0650 1.1725 1.8332 1.109 0.0975 1.4484 2.3718 1.058 0.1300 1.7240 2.9565 1.011 0.1625 1.9997 3.5943 0.964 * For crown deflection. 86 ‘y/W/ \l (a) Deck Bridge (b) Half Through Bridge m all Hm (c) Through Bridge Figure 1-1. Types of Arch Bridges. 87 Figure 2-1. Beam Subjected to Combined Axial and Lateral Load. 88 Y A (a) X-Y Coordinates. 83.). a V“ u). 21 (b) X-Z Coordinates. Figure 2-2. Coordinate System for Arch Bridge. 89 WE! TI 1 wfi Lil i 11L) L X (a) Combination of Dead Load W’ and Additional Vertical Load wa. f 5111177111} Fri; (b) Combination of Dead Load wk and Additional Longitudinal Wind Load we. (C) Combination of Dead Load Wf and Lateral Wind Load wa. Figure 2-3. Loading Conditions. 90 (a) Anti-symmetric In-plane Buckling Mode. K (b) Symmetric Out-of-plane Buckling Mode. Figure 2-4. Typical Buckling Modes. 91 _k r {1. Figure 2-5. End Displacement of Three Dimensional Beam Element. ' "Tn‘ w." Figure 2-6. Cross-section of Beam Element. 92 rib deck (a) Deck Situated Below Ribs. deck rib (b) Deck Situated Above Ribs. Figure 2-7. Tilted Load Effect. 93 Amnflmv .um mm u mcaommm hmwa.o n cmmm\mmflm u comm .uw oo.mm¢ .omoflum mmmm «0 “N? G! msowmcmEflo Hamumbo 9 % .Hum musmwm 94 _‘r 1? fl .5 I E o —-3/4 in. I : m lacing i E . .. g u c . i I l: .5 z _____. ___JL..1____..____—LJ! . I :6 g I I 0‘ I : : . g I a : i .H . I I o I ‘2 z ' 2 1135 41 x 7/8 in. 4 L'S 9 x 4 x 1 in. Ribs Properties Bracing Properties A = 1.6736 ft.i A = 0.8316 ft.i xxx = 13.8395 ft.4 Ixx = 7.9965 ft.4 I z = 2.1956 ft.4 I z = 1.4275 ft.4 KT = 5.9880 ft. KT = 1.9759 ft. E = 0.4175 x 107 ksf; u = 0.3; Cy = 40 ksf Figure 3-2. Cross-sectional and Material Properties of 95 Transverse Deck x-truss Bracing bracing Tower Continuous 4 ’1 ‘Beam I“l v{iiliiill-E‘. iiiflg.!22§§rt.-<< ‘f‘I'Illiggé‘ag!zfiiiéfi!i=i=:>\\ Column ‘ V’, Rib A \ ’\ X-bracing Rib cross-beam bracing 'I Rib Figure 3-3. Model for MCSCB Bridge. 96 (3) ,—- (4) ’9' (2) "9' (1) (a) Front View of Four Panel Model. (8) <7) ,. <9) - -‘ XX Ii 2‘. (b) Plan for Deck and Towers, §§§ (6)Egg (5) (c) Plan for Ribs x-truss Bracing, .b' (d) Front View of Eight Panels Model. Figure 3-4. 4- and 8-Panel Models for MCSCB Bridge. 97 Figure 3-5. Local x and Z Coordinates for Truss and Beam Members. Deck Slab t t4 D = 9 ft. B = 2.92 ft. C = 23.0 ft. t1 = 15/16 in. t 2 2 t3 = 3 in. t4 = t5 = 0.073 in. Figure 3-6. Three Cell Box Model for Torsional Stiffness of Deck Beams. 98 I‘ 6 ft. fip-I .\ °~ 60 '.'-"o.b'o."'° 9.“:5 030.0%) 0" 9.5- 71/4 00 . ....b.. in. C) F‘F"-lb in.__4_4 *9 —/l J T7/81’“ 5/16 inn-i It— Concrete Slab 52 in. Stringer-—————/////////’f e 1 [ IT7/8 in. H , 3'1 12 in. (a) Composite Beam Deck Slab and Stringer of Actual Bridge. S=28' (b) Spacing Between Composite Beams. Figure 3-7. Components of Actual Bridge Deck. 99 1 \\\\\\ (a) Shear Deformation of Slab. 'k FA“ * Ad 4 I (b) Shear Deformation in X-truss Bracing Figure 3-8. Modeling of Deck Slab. 1.0 100 (a) K-truss Bracing in Actual Bridge. 0.5 (b) x-truss Bracing in Bridge Model. Figure 3-9. Modelling of K-truss Bracing Between Ribs. 101 1.0 ML 777-..- Figure 3-10. Sketch of Actual Tower. 102 .mwpwum mfimh mo mcowusHom moam> :mmwm was Egannwawsvm Al 1. \‘4I. .Vk..r4\3d ‘4. ucmcwEHmuma can ucmsmomHmmwo Hmowuum> .ema ooom ooom ooo6 ooom ooom coca as x III m.mw m.mm H.om s.Hv m.mm H.m~ n.8H m.» 0- lb b L b b a 1 J 1 ID by d L . Ema V 1‘ V 1 \\ > U magnumcs .IIJ \x\\\ . mo.an.3\.3 \\ 019mm h b ’ D 4 ‘ ‘ d 1 ‘ .Haum 6855mm .HN. Lam. .av. .Hm. o.H H.H PPOT PIQTA 8901 pannqrnsm Ktmonun “1901:1811. 103 I 112;:>/’i,112.5 : 112.5 | Z .11112-5 All Dimensions are in ft. For Equilibrium Solution, P and Q are Load Increments; P = 112.5 Kips, and Q = 0.001 P. For Eigenvalue Solutions P = 1.0 Kips, and Q = 0.0. Cross-sectional Properties; _ 2 _ 4 _ 4 A - 2.7 ft. ' IXX - 3205 ft. I 122 — 4.45 ft. I and KT = 5.785 ft.4 Material Properties; E = 0.4176 x 107 st, u= 0.3, and CY = 40 st. Figure 3-12. Dimensions and PrOperties of Arch-A. 104' CH x.l. .cmmflm cam Esfiunwflwovm ucmcflsumuwa 0cm ucwEmomammfla Hummumq o.om H.Hh ~.~m . m.mm 1 1- an «L .21.. 8.33.".— ‘I \\ xnnnnr magnumca ‘1 fiP .....wa III o ".5qu A LrN O OH :N.NH ;~.¢H .m.wH P901 PTGTK peoq paqnqrzqsrq Ktuuogrun “‘[EDIQJQAu 105 P 1 P 11 . p 1 11 1 h 200.0 150.0 150. .....— .JL A»... 4 (2) (4) (6) (8) (9) (10) (11) 140.0 4 J: (1) (3) (5) (7) F" 400.0 4‘1 (a) Example One. A 1 I 80.0 ans iv F: 400.0 ,1 (b) Example Two. E = 0.4176 x 107 Ref. u’= 0.3 Figure 3-14. Arch Bridges Considered by Ostlund. '106 .mum3oa Emomlmmsue can mmsua no mofiuummoum .manm wusowm II [III 6cm .mmx bad x thv.o II E-I a: v um mc o w an HO C II H II H v um 0 com .u .m N m c umwwunwmonm Hmcofiuommlmmouo ocfi3oHHom on» cufl3 mucwfimam Emma mum mcfisaou II ‘13 .um o.m u some Hmcowuomm Immouo muamfiwam mmaum mum mumnEmE mmouo can x H0309 Emmm was mmaue Nona.“ O.m " wflmhw HMCOfiUmeImmOHU HHM “0309 mmsue O’SZ 0°SZ O'SZ O’SZ 0°52 0°52 107 v I IIIIIII.IIII'I-I'II. DIIII I I7! '1 III I . ' .III."ulll..‘II OI: » .III'IIO. II. I...» ..II..'-IIAII1 .Hw3oa mmSHBIEmmm o no mommnm pmaxosm .malm madman wvaz Uflovwm @002 HmHflh mmmam pmaxosnca AT sl..o. wmwfim Irina. _ _ wwaUSm 2+ :1? _ _ 9 1. v— n— (a) (b) (c) (d) . 108 £2222333 K-truss Bracing , AOAOAOAOAOAOAQA . Diagonal Bracing :IJ 11H Transverse Bracing Figure 4-1. Patterns of Rib Bracing. 109 .Ampoz umufihv mcwomum mmuuulx .mnflm momoz mo mmmsm Umaxosm .muv ousmwm AHMHQ ”H 8888 8 cowum>mam 08.85 80303:: I .0. I .0. 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I. 149 OH x m ucmfiwomammfio Hmowuuw> wan mmumumq N.Hw b u m.Hm b h I II ucwsmomammfia Hummus 1|! ¢.Nv .csouo um unwEmomHmmwo Hmuwumq can Havauum> mo :oflunaom Ezwunfiaflsvm comm mmwwhm noud m.m~ H.¢H P p 1 u u m.Nm # 1 db 1P ‘b } I ucmsmomHmmfla Hmowuuw> .11. hmwo.o .Hlm muzmwm as“! ..1 h.v ..mm.oa 1.5m.om oo.am .mm.H¢ :bm.Hm :oo.~w :qm.mh .hm.~m A .oo.mm .mm.moH chm.MHH 11 '[PJBZL'E'I 11 AnyaoIeA qdm 001 :9 peoqggugm P901 TEUOIQIPPV WIOJTUH 150 .wmonun :0 mucfiom mwuns Mom ucmEmomammwn Hmuwqu mo :oflunaom Enfiunfiaflsvm comm mmwfium nou< moH x ucmfimomHQMfln Hummqu n c.0h N.Hm m.Hm «.mv . m.Nm m.mm H.¢H . h.v ’ d 11: ‘D 1 1 P + 11 db <1 .0 can .n .0 mo mcoflumooH Mom H1m wunmam mom hvmm.o .m1m wonngn 001M KQEDOIGA ufim 001 29 p201 purm peoq Ieuoynyppv mzogtun “1919491“ EM .czouo um ucmfimomammfio Hazavnuwocoq.vcm Hmowuum> mo cowuaflom Esfiunwafiswm .mlm wunmwm x comm mOCwum non< MOH unmEmomammfio Hmowunw> mam Hmcfiwsufimcoq “I can 151 db 4 q 1 4. J1. mmm~.o u a : ucmswomammfio Hmcwcnuwacoq mmmv.o u a ucmfiwomamwfio chnonunocon ; mmvo.o u a ; ucmfimomammwo Hmcfiwnuwmcon : mmvo.o u a 4 ucmEmomammflQ HMOfluHm> t ootM A210018A udm OOI :2 P901 PUIM P901 Ieuoratppv mxogrun uTPUTPUQI5U°1u 152 w [:51'_1F——T"'1 ‘ W L- (a) Loading Condition Corresponding to Anti-Symmetric In-Plane Buckling Mode. wa 1 (b) Loading Condition Corresponding to'Symmetric In-Plane Buckling Mode. “WIL I k—r—T—r—‘IW X (C) Loading Condition Corresponding to Anti-Symmetric In-Plane Buckling Mode, Non-Symmetric loading. Figure 5-4. Loading Conditions CorreSponding to Symmetric and Anti-Symmetric In-Plane Buckling Modes. 153 .mawomoq Ego-«Joacoz Mom nongomnom Hanan-Hos, mo coma-H0950 .m1m ousmwm A.um. noncommom Hobauuo> m.m o.m m.~ o.~ m.H o.H m.o .171 “I n u u u n .V .H .0. \\ .ibLH M_M \\ o \ __ \ 1v N00 Aguouuwm mcwomoq How m nvlm whom-Tm ommv &\ u \ - s \ . m 0 8W 53- m n \ Mm. \ 1v "to ..m M \ H. oncommom \ \ Mm. fiGflHfiHQc—fi |'\ -. 1 PW \ ‘11. oncoonom ‘1' oncommom : m6 \\1 Hmocfiacoz Hmocflq M \o\ - P % o.o 10. REFERENCES Ojalvo, M., and Newman, M.. ”Buckling of Naturally Curved and Twisted Beams," Journal of the Engineering Mechanics Division, ASCE. Vol. 94, EMS. 1968. PP. 1067-1087. Ojalvo, M., Demuts, 2., and Tokarz, F. J., "Out-ot-plane Buckling of Curved Members," Journal of the Structural Division, ASCE. Vol. 95. ST10. 1969. pp. 2305-2316. . Wen, R. K., and Lange, J.. "Curved Beam Element for Arch Buckling Analysis," Journal of the Structural Division, ASCE, Vol. 107, ST11, 1981. pp. 2053-2069. Tokarz, F. J., "Experimental Study of Lateral Buckling of Arches," Journal of the Structural Division, ASCE. Vol. 97, ST2. 1971, pp. 545-559. Tokarz, F. J., and Sandhu, R. 8., "Lateral Torsional Buckling of Parabolic Arches," Journal of the Structural Division, ASCB, Vol. 98, ST5, 1972. pp. 1161-1179. Donald, P. T. A., and Godden, W. 6., "A Numerical Solution to the Curved Beam Problem," The Structural Engineer. Vol. 41, No. 6. 1963. Godden, W. 6., "The Lateral Buckling of Tied Arch," Proceedings of the Institution of Civil Engineers, Vol. 93, No. 3, 1954. PP. 496-514. Godden, W. 6., and Thompson, J. 6., "An Experimental Study of a Model Tied-Arch Bridge," Proceedings of the Institution of Civil Engineers, Vol. 14, 1959, pp. 383-394. Shukla, S. N., and Ojalvo, M., "Lateral Buckling of Parabolic Arches with Tilting Loads," Journal of the Structural Division, ASCE, Vol. 97. ST6, 1971, pp. 1763-1773. Johnston. B. G. (editor), Guide to Stability Design Criteria for Metal Structures, Structural Stability Research Council. Third Edition, John Wiley and Sons. N.Y.. 1976. .154 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 155 Nettleton, D. A., and Torkelson, J. 8., "Arch Bridges," Bridge Division, Office of Engineering, Federal Highway Administration, U.S. Department of Transportation, Washington, D.C. Bleich, F., "Buckling Strength of Metal Strucutres," McGraw-Hill Book Co., New York, 1955. Ostlund, 8., "Lateral Stability of Bridge Arches Braced with Transverse Bars," Transactions of Royal Institute of Technology, Stockholm, Sweden, No. 84, 1954. Wastlund, 6., "Stability Problems of Compressed Steel Members and Arch Bridges," Journal of the Structural Division, ASCE, Vol. 86, ST6, 1960, pp. 47-71. Almeida, N. F., "Lateral Buckling of Twin Arch Ribs with Transverse Bars," Ph.D. Dissertation, Ohio State University, 1970. Sakimoto, T., and Namita, Y., "Out-of-plane Buckling of Solid Rib Arches Braced with Transverse Bars," Proceedings of the Japan Society of Civil Engineers, No. 191, 1971, pp. 109-116. Sakimoto, T., and Komatsu, 5., "Ultimate Strength of Arches with Bracing Systems," Journal of the Structural Division, ASCE, 8T5, May 1982, pp. 1064-1076. Sakimoto, T., and Komatsu, 5., "Ultimate Strength Formula for Steel Arches," Journal of the Structural Division, ASCE, Mar. 1983, pp. 613-627. Yabuki, T., and Vinnakota, S., "Stability of Steel Arch Bridges in State-of-the-Art Report," Unpublished Paper. Merritt, F. 3., "Structural Steel Designers' Handbook," Chapter 13, McGraw-Hill Book Co., New York, 1972. Lange, J. 8., "Elastic Buckling of Arches by Finite Element Method," Ph.D. Thesis, Michigan State University, 1980. Rahimzadeh-Hanachi, J., "Nonlinear Elastic Frame Analysis by Finite Element," Ph.D. Thesis, Michigan State University, 1981. Wen, R. K., and Rahimzadeh-Hanchi, J., "Nonlinear Elastic Frame Analysis by Finite Element," Journal of the Structural Division, ASCE, August, 1983. Bathe, K. J., and Wilson, E. L., "Numerical Methods in Finite Element Analysis," Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976. 25. 26. 27. 28. 156 Mallet, R. 3., and Marcal, P. V., "Finite Element Analysis of Nonlinear Structures," Journal of the Structural Division, ASCE, Vol. 94, No. 8T9, Proc. Paper 6115, Sept. 1968. PP. 2081-2105. Cook, R. D., "Concepts and Applications of Finite Element Analysis," John Wiley and Sons, Inc., New York, N.Y., 1974. Timoshenko, S. P., and Gere, J. M., ”Theory of Elastic Stability," McGraw-Hill Book Co., New York, N.Y., 1961. Gere, J. M., and Weaver, W. J., "Analysis of Framed Structures," D. Van Nostrand Co., 1965. APPENDICES APPENDIX A LINEAR AND FIRST ORDER NONLINEAR STIFFNESS MATRICES OF A TRUSS ELEMENT For the linear and first order nonlinear stiffness matrices of a truss element, the twelve degrees of freedom defined previously for the beam element (see Figures 2-5) are used here also. Note that only non-zero elements are given below. Linear Stiffness Matrix k(1,1) = k(7,7) = 3.. IL k(7,1) = k<1.7) = ‘ifi. First Order Nonlinear Stiffness Matrix n1 (2:2) = n- (313) = n1 (8'8) = :11 (2,8) = n1 (8'2) = N“. (3'9) = in which v — E (u' - u) 2 2 R v I and u2, u1 are axial displacements of coordinates. 157 :11 (9,9) n1 (9,3) the ends in local APPENDIX 8 PROGRAM NEAMAH .Bil__D2fiQLiRLiQn_Q£_SnDLQnt1n£a. Program NEAMAH, has been described in Chapter III, and a listing of the program is given at the end of the section. The input data is explained with enough comment statements as they enter into the progranu Other comment statements are used as needed. In the following, a brief description of the subroutines is given. The main program (EIGEQDK) directs the flow of execu- tion by calling the appropriate subroutines for each step of the solution procedure. Subroutine NODDATA reads the data of the structure geometry which includes mainly the coordi- nates and degrees of freedom of the nodes. Subroutine ELEMENT calls the appropriate subroutine (BEAM or TRUSS) to read the elements properties data. Subroutine BAND computes the semibandwidth, MBAND, of the structural stiffness ma- trix. Subroutine BEAM and TRUSS evaluate the linear stiffness of the beam and truss elements, respectively; Subroutine TRANSFORM and INVTRNS are used for geometric transformation from local coordinates to global coordinates and vice versa. Subroutine SBEAMl, SBEAMZ, and KEPSIOl, respectively, eval- uate the non-zero entries of [n1], [n2], and [K o]. The 158 159 assembly of [k], [n1], [n2], and [K,o] into the appropriate global stiffness is accomplished with subroutine ASEMBLE. Subroutine LINSOLN solves the system linear equation by Gauss elimination. Subroutine STCONDN condenses the struc- tural linear stiffness matrix and load vector into the degrees of freedom which have been established in subroutine NODDATA. Subroutine RECOVER recovers the internal degrees of freedom of the structure after using subroutine LINSOLN. Subroutine IDENT identifies the displacements obtained from LINSOLN with the nodal displacements similar to those found in the recovery process. Subrutine DECK was developed for the inclusion of the rigid deck effect or the "tilted load effect." Subroutine LINDECK and NONDECK included approximate deck effect "tilted load effect,“ assuming that the deck is flexible. The approximate effect of a tower was included in subroutine TOWER. The linear eigenvalue and the multi-eigenvalue solu- tions are obtained using program EIGENVL. The nonlinear eigenvalue solution is obtained using NLEIGNP. 'For the solution of the quadratic problem, subroutine NLEIGNP uses the modified regula falsi method of iteration by calling subroutine MRGFLS and the function subprogram DET. Function subprogram DETl evaluates the determinant of the structural tangent stiffness matrix. Subroutines ENDFORC evaluates the element end forces. 160 E 2 H . .1 H i . I] C | E . The variable names used in the program are listed below in alphabetical order: ELQQan_NEAMAB- A(M) = The cross-sectional area of element M; A701D(M), A7TOT(M) = Parameters related to element M for evaluation of the initial strain stiffness matrix; BOL(M,J), BTONM,J), BE(J) = Intermediate parameters for the evaluation of initial strain stiffness matrix; D(I) = Displacement vector, found from the solution of the system S*D=R. I varies from 1 to NEQ; DTOT(I), DACTUAL(I) = The same as D(I) but for total dis- placement measured with reference to the beginning of each load increment or initial geometry, respectively; DETER, DETERMNT = Determinant of the structural secant or tangent stiffness matrices; DETCHK = Constant always zero, so the program will stop if the detrminant increases as an indication of instabil- itY: DN(I,1) = End forces in global coordinates for each element. I varies from 1 to 6; E(N).= Modulus of elasticity of element group N; 161 ES(I,M) = End forces in local coordinates for element M. I varies from 1 to 3; G(N) = Shear modulus of element group N; IA(N,I) = "Boundary condition code" of node N for its Ith degree of freedom. Initially it is defined as follows: IA(N,I) 1 if constrained; = 0 if free After processing, IA(N,I) o if initially = 1, equation number for the DJLF. if initially = O; IB(N,I) = "Additional boundary condition code." IB(N,I) 0 if free N if slave to node N; -1 if to be condensed. After processing, IB(N,I) is unchanged except, IB(N,I) = -(condensation number of the D.O.F. if initially IB(N,I) = -1)3 ICALl, ICALZ, ICAL3 = Variables controlling print-out (more details are indicated by "comment statement" in the listing of programs): ICAL4, ICALS, ICALG = Similar to above, use for approximate in-plane and out-of-plane deck and tower effect only. 162 ICHECK = Parameter used for Newton-Raphson approach in La- grangian coordinates to control the type of computa- tion needed in each load increment; IDET = Parameter used for evaluation of the determinant of the secant or tangent stiffness matrices either before or after Gauss elimination process; IGOPTIN = Paramether used to specify type of the geometry for plane frames (idh, circular, parabolic arch or arbitrary geometry): IPAR = variable identifying appropriate "Tape” for storage of different structural stiffness matrices (i.eu. [K], [K 01' [N1]: [N2])7 ISTRESS = If E0. 1, compute nodal forces and stresses in the structure. If E0. 0, skip; IXX(M) = Moment of inertia about the x-axis of the cross section of element M; IZZ(M) = Moment of inertia about the z-axis of the cross section of element M; KT(M) = Torsion constant of element M; L(N,K) = Variable identifying the Kth element in the element group N; LE(M) = Length of element M; 163 LODPONl = The degree of freedom at which the load to be increased (automatically generated by the program with proper use of LOADDIR. If the load to be incremented is not at the first node, then LODPONl is to be inputted manualy by changing ITETO value to 0); LNODEl = The node at which the load to be incremented, see previous note; LDOFl = The local degree of freedom of the incremented load at the previously indicated node; MBAND = Semibandwidth of structural stiffness matrix; NCOND = Total number of degrees of freedom to be condensed out; NCOUNT = The order of load increment in incremental approaches; NE = Total number of elements in the structure; NEQ = Total number of equations; NODEI(M) = Variable identifying the number of node I of element M; NODEJ(M) = Variable identifying the number of node J of element M; NSIZE a Total number of degrees of freedom, condensed and free, of the system. (NSIZE = NEQ + NCOND); 164 NUMEG = Total number of element groups; NUMEL(I) = Total number of elements in element group I (in NEAMAH); NUMITER = Number of iterations at each stage of computation; NUMNP = Total number of nodal points; PINT(N,I) = Initial load applied at node N, in the Ith direction. PINC(N,I) Load increment at node N, in the Ith direction. PTOT(N,I) Total load at node N, in the Ith direction. PACTUAL(I) = Applied load related to the Ith D.O.F. in the structural load vector at each stage; PSAVE(I) = Initial reference load in the Ith direction, initially equal to zero; PSTART(I) = Initial load in the Ith direction; PTEMP(I) = Resistance in the Ith direction, with reference to current updated state; R(I) = Load vector of the system; ROT(I,J), ROTRAN(I,J) = Rotation and inverse rotation matrix for each element (I s 1, 6, J = 1, 6), respectively; S(I,J) = Tangent stiffness matrix of the system; 165 SCALE = Scale factor in the evaluation of the determinant of the structural stiffness matrix; SE(I,J), SEI(I,J), SE2(I,J) = Element stiffness matrices (i.e., [k], [n1], [n2], respectively); ULOC(M,I) Identifies local. displacement :hi the {Ith direciton of element M (I varies from 1 to 12 for three dimensional case and from 1 to 6 for two dimensional);- W(I,J), WCHK(I,J) = Incremental recovered displacements (used In iterative process) related to node I in the Jth direction; WTOT(I,J) = The same as W(I,J) but for total displacements; X(N), Y(N), Z(N) = Global X, Y, Z—coordinates of node N; ZPGM(M) = Rotation of local major principal axis. SDBBQQIINE_IBANSEM Rcol(I) = Identifies the entries of rotation matrix for three dimensional beam element. I varies from 1 to 9; SUBBQHIINE_IN¥IBNS V(NP,I) = Identifies the element local displacements for nodal point NP and Ith direction (I varies from 1 to 6); 166 SHBBQHIINE_§I§NDN RC(I) = Condensed structural load vector (I = l, NEQ); SC(IpJ) = Condensed structure linear tangent stiffness matrix; 3flBBOHIINE_EISENYL_1E1§ENL_IDAIAl EIGEN = Eigenvalue; EIGNVTR = Eigenvector corresponding to EIGEN; EPSI = Tolerance; MAX = Maximum number of iterations allowed; RHO = Rayleigh quotient; KB 2 Vector that stores the approximation to the eigenvector after each iteration; EQBBQHIINE_ENDEQRQ DN(I) = Stress resultants on the nodes of each element; SUBBQHIINE_NLBIGN£ A,B 2 Variables defining the interval in which the eigen- value is enclosed; ERROR = Upper bound on the computation of the eigenvalue after convergence; FL 8 FTOL 167 ‘Value of the determinant of the matrix:S = K + L*N1 + L*L*N2 at the converged value of the eigenvalue; a Convergence criterion for sufficiently small value of the determinant of eigenvalue; L = Converged value of the eigenvalue; L = (A + B)/2; NTOL = Maximum number of iterations allowed; XTOL = Tolerance; SUBBQQIINE_MBGELS IFLAG = Variable defining the status of the iteration. If FA = FB E0. 1, convergence was successful. If E0. 2, no convergence after NTOL iterations. If E0. 3, both endpoints, A,B, are on the same side of the root, hence method of iteration cannot be used; Value of the determinant of matrix S at interval endpoint A; Value of the determinant of matrix S at interval endpoint B; W = Weighted values of the root between interval endpoints A FW = and B; Value of the determinant of matrix S at the weighted value W; 168 EHNQIIQN_DEI DET = Value of the determinant of the matrix S = K + L*Nl + L*L*N2 at a particular value of L; K(I,J) = Part of element S(I,J) corresponding to linear stiffness K(I,J); L = Load parameter; N1(I,J) 8 Part of element S(I,J) corresponding to matrix N1(IIJ); N2(I,J) = Part of element S(I,J) corresponding to matrix N2(IIJ) o 1159 WNEAMAH (INPUT OUTPUT- 6s ATAPE60- PINPUT TAPE61- UTPUT ‘TAPE1 TAPE2 TAPE3 TAPE4T APES P§6T ,TAPEg TAPE,,TAPE10, +TAPE 1, TAPEiz, TAPE13, TAPE14 TAPEl TAPE16' TAPE 1 *****t****************t**i*fiit.***************************t**t******** PROGRAM NEAMAH THIS PROGRAM. E SOLUTION OF LINEAR AND NONLINEAR EQUI AL BUCKLING PROBLEMS OF ARCH BRIDGE ST N ELEMENT METHOD WITH NONLINEAR ELAS AND TRUSS ELEMENTS IN THREE DIMENSIONAL SP THE PR RAM CAN ALSO BE USED FOR SOLUTIONS OF OTHER EEREETU S MADE UP OF THESE ELEMENTS IN THREE DIMENSIONAL i***i*tt*ttttitti***********fl*********i**t*********i*.*t*******t***it REAL IXX,IYY,IZZ,KT,II,JJ,LE,NISTTOT COMMON/1/NE,NUMNP,NUMEG, LE(54) ,NUMEL(3),IPAR,ICAL1,ICAL2,ICAL3, 1 TRESS %6:Q18§g¥ 5¥3§"§b§,§I§§),2<30) COMMON/ MMO NO §I(54), ,NODEJ(54),A(54),IXI(54),KT(54), :GIII ”I GSIiSAIEG 311%3?,G4(168,10),RC(168), (Iififildbfifiiflwiififiiniflfin ififlfii$iiflbfifiiiflifififi + *"* UCKfiDU (NDOWMNmFKMWPFKND , 30 9,0212; R¢8i(9), MSUOPTN, NlGOPTN A70LD<54),EOL(54,5),ET0(54,5),EE(5) 12 A 8) EMP(1 §) PgTART(168) ,DTOT(168) Mg???” L? 25%3"%§~“?’“%As?6’151’333325" SRK(168 ) REPPTMP PROTYPE EIGVALU,PRIOPTN,DETOPTN DOF,TLDOF *t******it*iit*******t**i*********fi**§********************************* DATA FILES I7 FORK BEAM ' EMENT TAPES O 11, PO A TBUYS ELEMENT KNl STRUL.JRAL v fififiifi * TAPES §% 2 PORN . TAPES i4,1§,16 P0P2 KEPSIO ; t *****************t**************fi*********i**t*********************** nwssussws: 0dflfl 3’30“ ”W MN” MEAD :4» tit!ti*t*****i*i*************fi************i************************fi**** RECORD NO. ONE UMNP: UM IDATA A: 0 TO CHECK THE DATA IN JT 1 PROGRAM EXECUTION ICALl: RngAD VECTOR AND s. CTURAL PIINEAR STIFFNESS 0 BE PRINTED(ICAJL =1 RZDISPLIS MENT VECTOR TO BE PRINTED S R LINEAR STIFFNESS MATRIX IN LOCAL OR GLOBAL DI INATE TO BE PEIgIED' ,ALSO FOR DETAILS OF EIGENVALUE TERMEDIAT RESULTS IN SUBROUTINE DECK PRINTED NTERMEDIAT RESULTS IN SUBROUTINE LINDECK PRINTED ICAL6: 1 INTERMEDIAT RESULTS IN SUBROUTINE NONDECK PRINTED ti***t********i*****************i************fi************************ til-*filefiIifltfififiqtfififi fifislifiilfltiith‘fildOIfi 'nd, 170 READ(60 1010) TITLEI TITLE2 TITLE3, NE, NUMNP, NUMEG, IDATA, ICALl, + ICAL2,1CAL3,ICAL4,ICAL5,ICAL6 :i***t**t*******tii**i****i**t1*tfiiifittitkiiitti*ttiitiitttttti*fitttt*ii ; nCORD No. TwO ; . ENTEIS IN THIS REC RD 15 1 FOR AFFERMATIVE RESPONSE * ; ELSE INPUT UNLESS OTHERWISE INSTUCTED. : * PRIOPTszTERMEDIAT CALCULATION TO BE PRINTED. * * NZOPTIN:FOR N2 To BE USED * * NlOPTIN: FOR Nl TO BE USE * * ITERCHK: FOR ITERATION APPROACH. ' * MSUOPTNzl CONSTANT WRAGE STRAIN FOR EACH ELEMENT. * ; MSUOPTNzéL33§§%N Is QUADRATIC FUNCTION OF SLOPE AT EACH ; * NlGOPTN:O FOR LINEAR EIGENVALUE SOLUTION N1 IS USED * * :1 FOR LINEAR EIGENVALUE SOLUTION NISTAR IS USED * * IFIX :0 FOR UPDATED LAGRANGIAN APPROCH. * * :; FOR FIXED LAGRANGIAN APPROCH. ‘ ' JUSTK :1 ONLY UPDATED LINEAR STIFFNESS MATRIX USED ELSEIO * * TOLER :TOLERANCE FOR SUCCESSIW ITERAT:ON CONVERGENCE ONLY * * DETOPTN :0 NO CONTROL ON THE DETERMINANT OF THE T.STIFFNESS M. * * DETOPTN: l EXECUTION WOULD BE TERMINATED IF THE DETERMINANT IS ' * EQUAL TO ZERO OR THE VALUE OF THE DET. INCREASED. * *I‘t‘tt‘k‘kititijttii *‘ktiit'1‘i'kfiiittt*t*iii*i*ttt**ti*i**t*******i*i***ti READ(60, 6971 HI PTN, NZOPTIN NlOPT nRCNK, _ MSUOPTN' N COM WII JUSTK, TOLER, DETOm 6911 PORNAT<éI F10 :**iiI******tttt***iififiii'{it**ttt**ittlit*i*t*fifit*I*Ititiitfiittfiftitflf: RECORD NO. THREE : LLOWABLE TOLERANCE FOR FORCE COMPONENTS OF * ED FORCE VECTOR ' LLON ABL E TOLERANCE FOR MOMENT COMPONENTS OF * 559.5 95$§.VECT R I * : * DE * UN * E ’ N t * *******ifig R’fit *tt**iifliiiiIti’ttfiI**fi**i*i***i*t*tt*t W(ITER .E >READ(60,6948) DELTA1,DELTA2 6948 FORMAT(2 21.?5I t*tR*tiii***it*t**itttititfitittliifiiiiti*i*t******Cit.******t*t**tttfi**t : RECORD NO. FOUR : a a * PROTYPE:1 IS FOR INCREMENTAL LOADINGC IN FIXED COORDINA ES * * SECANT STIFFNESS APPROACH SUC CESIVE ITERATIONS * ‘ PROTYPE:2 IS FOR EIGENVALUE PROBLEM * * PROTYPE:3 FOR INCREMEN AL LOADING IN UPDATED-COORDINATES * * PROTYPE:3 FOR NEWTON-RAPH ON FIXED COORDINATE APPROAC * * PROTYPE: ITERCHK:0 & JUT :1 FOR THE UPDATED-COORDINA * * APPROACH BY UPDATING ONLY LINEAR STIFFNESS MATRIX * * WITH NO ITERATION. ‘ * EIGVALU:1 IS FOR LINEAR EIGE ALUE PROBLEM * * EIGVALU: IS FOR OUADRATIC EIGENVALUE PROBLEM * * EIGVALU: IS FOR NCREMENTAL LOADING IN FIXED COORDINATE? * * EIGVALU: FOR INCRE ENTAL LOADING IN UPDATED-COORDI NATES OR * * FIXED COORDINATES) * * ISTRESS:1 ELEMENT END FORCES SHOULD BE EVALUATED. V * * ISTRESS:? ELEMENT END FORCES SH OULDN, T BE EVALUATED * * EFFICIENT CO INC ‘ * IPART :1 IF CONV. OF DISPLASEMENTS AND INTERMEDIAT VALUES * ‘ OF DISPLACEMENTS IS TO BE PRINTED ELSE IPART-O * * (E?UIL.SOL.ONLY * * LOADDIRz- IF LOAD IS TO BE IC MENTED IN THE X DIR. * ' LOADDIR:0 IF LOAD IS TO BE INCREMENTED IN THE Y DIR. * * LOADDIR:l IF LOAD IS TO BE INCREMENTED IN THE DIR. * * IDECK :0 NO DECK INCLUDED. * * IDECK :l RIGED DECK. * * IDECK :2 INPLANE EFFECTO DECK. * * IDECK :3 OUT OF PLANE EFFECT OF DECK. * * IDECK :4 EFFECT OF DECK AND . * * IDECK :5 COMBINED OUT' OFF AND IN PLANE PLANE EFFECT OF DECK. * : ISHEAR :l SHEAR EFFECT IS INCLUDED ELSE 0 : *ttit.*tit***t§****R****I****fi**I.***************i**iifif*fiti*ttt***tifi** 1371 READ(60,1) PROTYPE, BIGVALU, ISTRESS, IPART, LOADDIR. IDECK, ISHEAR +W¥£3Eé61¢20gongE%E}chI§L%2TITLEB, NE, NUMNP, NUMEG, IDATA,ICAL1, wRITE(61 972 PRIOPTN NZOPTIN NIOPTIN ITERCHK, +MSUOPTN, fi GO Irxx, 305Tx ,DETOPTN, mLDR 6972 IES§W *EiOPT16/10331 iégfl gm ITERH :2/*19§281 §°§§§u§éT§£*.IZ/,IOX. 1 +103, QBETB$T§3{' §IXTOtzR-;m 510. g? Tt'* I C 3931 £5£§§¥§ox““:EfiépézY51E§isis/?8x’sBEEE?fl §$‘35,/) IWRITE(§1,87 5) PROTYPE,§ flISTfifiés IPART LOADDIR. IDBCR 8755 §§§:;§ gééés§g*§“°7fgs' {SHEAR_ M1; V§3§ '«téiébxa-a :2/. t****t***********t*it*tt!*i**fifi******fii**t******Q***************.**tfiifit RECORD FIVE * t 'k * IGOPTIN:1 roa IRCULAR ARCH IGOPTIN-Z r a pARADOLIC ; IGSPTINnO FOR OTHER GEQMETR as * *ttt****t*t:i92§*IH;§tf§9§§¢§*§9¥ it geztggg§§tt wgti*£§9§$£§:9&;*ttititttt: BEAN $1gbé§8§IN 287 PORMA }/ 88, 1'I &PTIN-*, IZ./) Iifiifl' ****.**t*t.*******t******************************tfittfli************tt*t : - “5552139 §1§----1--29§9£§§§--1- : * MAXITER MAx. NUMBER OF ITERATION r 3 EACH LOAD IN azuzuT . ; TLDOP nausea or TOTAL LOADS APgLIgD ON THE 5m wagons. ; * Fifi? fi:i .....ifi*iA£.Lo OOOOOSOOIIOOOO..0...OOIOIOOOOOOOOIOOOOOOI * * 9 NC N,I INCRBMENTAL & ADINO * g T N,I TOTAL LOADIN : : ”SDSW Phi: 12T°mpronx AND 2 CONCENTRATED roaczs : : ,5 ANDTg %'§ 2 MOMEfiTs : tittttttttittttttttitt*tttttitttttitttt******ttttttttttttitttt*ttt*tiit. c c 1408 ITER TLDOF szan TLDOF 4801MTsé ,12,1ox,*uuunzn OP LOADS -*,12) WRIT DO 4 407 :ZN’I‘gthl'4O I)8§§NC(:, I)'PTOT(N, I)-0. 0 406 ““¥:?19i&§§?85tpé¥3§2a% b?‘¥§ é?fi?83%5?3834?fi?85%> I F 3 +*PINTI§13,'*PIfi ( $8m HE \DUWKD MN E; FORMAT W'?& 65x *MAXITER -* 15) §' filgADINO CONDITIONS : *//.6x, *NODE* 7x,*DOF*,léx, 1o FORMAT 4x,15, 11x, 0T516. 4, 10:, r10. 4, 1oz, F10. 4) ***************************t**t******t******t**********t*t***********t. READ NODAL POINT DATA * ********t***********************************************************i*if .finwbubfiflfl> 1372 CA L NODDATA(I PTIN I PROTYPE.NE.§? 60 $0 3021 P fszkgiég§§fgo.-I)SCALE-10000.0 DE ggx- . DO 1 181 NE 5001 PSAVEQI)IDACTURL(I)-DTOT(I)-O.O :......9939¥9$£S.§§§§§9I£9§.9§.299§9§l.9¥9.E?§§§§€9§9£§9.9I91§I....1 IF LO DIR)1655,1665,1675 1555 méaz’i‘g IXAE¥36 - GO TO 685 1665 IHORZ- IVERT- 08‘350 685 1575 IHORZ-é IIIII: i685 CONTIN ITE 1 s IH R2 VERT LAT : 1686 ORMAT%‘ 5,902,91NO§z-*,I§.5x,*IvEET-*,Ia,sx.*ILAT-*,13//) ; IF LOAD NANTED FOR SPESIEIc LOAD LET ITETo-I ; CHOOSE THE APPROPRIATE VALUES OP LODPON1,LNODE1,AND LDOPI I To-o IgfxTETO 50.0) GO TO 9152 LODPON1-2 53325158 IF TO. .0 TO 700 9152 3§ g? ¥.¥‘,m"8 I IA N I I . TO 3 IP PINT(N,I§9E°20 TO $§8 IF x.E .1. NO 2'58 0 TO 300 IF IHO§2.EO. 2 TO 9 LDOP - GO TO 900 909 IP IVERT.B '8 AND.I.58 2)GO TO 300 IF IVERT.E . {00 TO 9 52059125“ "v LDOP - GO TO 900 499 IerLAT.E ‘8 AND I.§8 g)GO TO 1093 IP ILAT.E is? To 9 98%? :31 ”'1 LD F -; GO TO 0 1093 PRINT *,9 PROGRAM CAN OT CALCULATE THE VALUE OF LOD Ni“ PRINT * ' HELP wANTED, PROGRAM STOPPED AT APP. LINE 6 ' swag? 2§§ c NTINUE CONTIN WRITE( 2900)LODPON1 LNODE1,LDOF1 2900 +§g ; §°*A4/NOD§':T¥§ g.0‘siT§NDWgIgH*L?g? HAS EEEN INCREASED-*, I g.i'gmp I v I 0 o v 3010 g fiégi-0.0 ' 2 UPDATE ORIENTATION OP PRINCIPAL AXES AND NODE COORDINATES 001 IE ICHECK E 1 60 T 21 1‘55 £656‘afiéméé" 9°” I-NODBI(M) ' 3020 C 3025 4995 3882 4998 §§§§ :i*ttifffl*fit*itt.filt.*filfi**fi§tttiQR!ifltfitiitttttiflifii*tttittttttiittiiti RE : i 1 i t t 5927 222 444 3916 3917 5341 173 J-NO (M) XI-x I YI-Y I 21-2 I XJ-x J YJ-Y J za-z J EL- 2§T(( -xI)**2+(IJ II)**2+(za-2I)**2) cx- -XI /EL CY- YJ-YI /EL cz- zJ-zg EL +321FA'0' ( W(I 4)+W(J 4))*cx+(w(I 5)*W(J 5))*c2+(w(I 6)+w(a 6))* 2PGM(M)-2PGM(M)+DALFA D 02 -1 NUMNP x I -x I +w I,1 Y I =1 I +w I,2 z I =2 1 *w I 3 IF PRIOPTN.E§. ) GO TO 3021 wRITE( 1,4995 FORMAT * 5 // 10x, 'NODE‘, lox, 52(1)*,10x,*y(1)*,lox,*z(1)*,/) 381129 1'19g9MNI x( ) ,z(I) FORMAT é ‘,/,léx ,15, 3Fi5. 91' CONTINUE wRITE( 1,49981 SgRggT ’7' /& ox, *MEMBER*,1OX,*ALPHA*,/) WRITE? 1,5995?& 2 M(Ig FORMAT * *,/. a) CONTINUE CONTINUE AD AND STORE ELEMENT DATA ALL LEMENT PROPERTIES WILL BE READ AT THIS STAGE AS FOLLOWS: .BEAM ELEMENT 2. TRUSS ELEMENT I? ONLY TRUSS ELEMENTS ARE USED LET NUMEL(N)-O FOR BEAM ELEM. lifii" itfiitfii**tfifi**t*itittitti*t****.*tl**Iifiittit!!!Ifitfiitiltttt*ttttfiifitfi IPéR-NUMITER-l N FNPROTYPE. E1?) CKb 3° 0 1%?Dsifd1N1EéQ $1EL 2Nm%3927 IF PROTYPE. NE Q3 ICHECK EALLE ,ICHECK, PROTYPE) N.E GO TO 5928 ( O 9N9:EG) ATA. E0. 1) GO TO 900 TE§?R§FI?’ 88 % 33%? mmémme‘zmmemz 5991959.§$£§§§§§§.¥92§£¥............ 16L(N1;EQ .51 GO TO 3916 MT *119/ %{{:me *SEMIBANDWIDTH -* ,13) s N.EO GO TO 211 MEOEIPEN 0211 L(NN). ENUM1? GO TO 5745 E 9”” O z,.~.«z utnaévu H6 : E§:wa ngfiA 33:“ ans: Hug (“was : nhoa HrunHm u > O>”fl8 E .1) A7OLD(M)-0. 0 om mEo. 2) BOL(M,I)-0.0 BQSQ¥8§§8233§ 174 .PSAVE(LODPON1).EQ.0.) GO TO 5010 NITIAL LOADS AND NODAL LOADS INTO LOAD VECTOR matgttéggégtmtEgg?!ttitittttttttiittitttttttt Q E N ) I ES 61 1 P 1 2 7 0 31 - W R7, 0 88 9 .5 l 31 0 3 )I 66 l 88 6 4 0 32 2 2 t I 33 O 2 99 7 9 5 4 4 .1: 233 R 44 44 00 I .33 PT. Ox! m 0 My. TT 0 N T( K I 8 TE 00 T T RT CNN KLTT P TT 00 ADV; H CL “V. W O 0 GO 0 \I 0 TA5 EAOO T 00 G G R a G I G ST . COO ED CGG O GG G A . t I P50 IGG ZN C E 51R ED G) RE a) W TR 5* ) ) O. P ) N ) 9P1 1A 1) )2 P \IZN OZ 0 SA ... M 3) E 0 ( 0 (F \1 5m .0161 . I \IOIA .) .IA 1 ) 0A 1T ... ( .1 N . T . X I )1 N N .3 .SW 8 3 IS 8 258 NON 2 EB NS IS. E. .0 Q N OTO)X 11. II (E.ON E( .ON 3( .NM .E( . NM“ R1 v... ENQ1.N6E 1 E1128 ..QllMNQEII I1 OEII I2 QII 1.19 I M 3* L .E ..0 II.P .2A2I 8...- .E11 2.11 Ell .0E11M} L a E E.21- 11) x 444: EE. 1J0 KIN-.0 IN .NI-O uN ...0 NNI . a.) 1- .. M PKl ) ._ .1 \I 0 I . .K I HKIIJ . N( KIIJ . N( KIJ . (S K 1.01.11: E At E vF¥1 I NI.I JEEP1$1$ “Hr. nu CCT nu .LE 6 r¥1 0 .LEE r. 0 TJLPEP. 11:24 W a S TE23( N (UUO; E 33.LRW..P11- 1 B EOE-.1. 2 8 E66. 3 EWE 11 .720 I ST: ANOH 0T 5? 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OX,*PAW P1”: 1 x119.4{/ N -* D 8;- {OTU§%§:&0PON1)§NSSEEEES(P§0T?LNOBD§ mti? ) so TO 900 AC §UHUHH§ 040% mun-mm Hnomhewaxana \A+QW(>~ our4mn ”“ ~nnmx IV 'UUfl' P5203: H0O? ”PW- 88?888283¥8z38moem8m “A 2290A HH gay 4 Hfi‘b .831)WGOGTO 2118 7). 80. .0G GO To 5281 -1m A QB 1) A7OLD(M)-A7TOT(M) O. 2) BOL(M,I)-BTO(M,I) cnguug A!" m: 2" a dbde éééa ONO- VH.. :Igg‘g ‘355 1,§unnp }:BQ.O) GO To 451 +PINC(N,I) I';'Nggéanr I -R I .829 p TART‘Ig-REI;+PSAVE(I) gTN gg, 50) GO TO 6977 ,,*R(I)* /) is: Li‘éfia'é) N ( OmQH I"! HHHZ x «we "”38 mm me'fi'flNHg I H7010! 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IA ,1 , 3x , ax,4x,1ay, 23T1,3§,3HT§,3x,2HT N H g H 66" ’fiiaHZ *¥¥¥+iddfl rv*4 Aflv<) M4 q~xbewanhnutH IH rhea N MM“ wa\loz WWN Egg 1. ~DH a ) X,4HNEQ-,13,33,6HNCONDS,I3) 3 SUBROUTINE ELEMENT (N,ICHECK,PROTYPE) ttractitatqtattit*atttwtttttatittttatttttwtattttccttttttttatattnag ...IQ.§§E¥*¥§§.§§§§95§3§2§.§§§¥§§$. gggggrgggltittttkfiittfittittifit +€g¥§g§ 1/NE,NUMNP,NUMEG,LE(54),NUMEL(3),IPAR,ICAL1,ICAL2,ICAL3, INTEGER PROTYPE IF "'28'%; ALL BEAM(N ICHECK paowyng g; N E . ALL TRUSS(fi,ICHECk,PROTYP ) sun SUBROUTINE BAND ta«atatttttrattaaat:t*tatttttcatn*tittttttt*tttttattttttattttttatt TO COHPUTE SEMIBANDWIDTH OF STRUCTURE STIFFNESS MATRIX DONE BY FINDING THE MAXIMUM DIFFERENCE BETWEEN THE E UATION NUMBERS ASSOTIATED WITH THE NODES OF A P RTISULAR ELEMENT tttitfiti it************if******I**t!*t*t*fi*i***i**t*fi******fi*§*flii 800 t 300 t 700 185 *gg¥gggél/NE NUMNP, NUMEG, LE(54) ,NUMEL(3),IPAR,ICAL1,1CAL2,ICAL3, COMMON 2 N5 QfN OND AND EN COMMON5371§ fiEBN (30E go ,2 go C?MMgN{ {E ii pN?§E¢ §,NO E3 4 ,A 4 ,Ixx(54),NT(54), +L 3 4 22 DIMENSION 11 DO 100 N-I Nm 13 IE(NUMEL( N Q.) GO TO 100 NAME- L N WRI 32?? IN NUMEO, NUMEL(N), NAME DO N-1,NAME M-L N K) NI-NODEI M; NJ-NO 53 M MIN - 0 MAX - IFEN.E .1: IN-g IF N E IN- WRITE(61§18 )N,NI,NJ 1N,L(N K),M DO 0 1- IN II?I?-IA%NI I) II I+IN)-IAINJ,I) CONTINUE 115-33” 1 N I WRITE96i,103§I 11(1) CONTINUE DO 700 I-I,NI 1E 1151;.NE.8.AND.IIEI .GE MAX MAx-II I IF {I I NE. .11 I .LE MIN MIN-II I ‘WRITE( 104)I,II(I ,MAX,H N CONTINUE MMBAND-MAX-MIN’I MMBA ND. E. MBAND)MBAND-MMBAND IF( WRITE(61,105 MAX, MIN MMBAND, MBAND CONTINU CONTINUE RETURN ‘ EQZZET(*1*,1,103,*N-* 15, 5x, *NUMEs-*, 15, 5x, *NUMEL(N)-*, 15, 5x *NAME’ ’EO T t * 10x *R- 15 ,*NI-*, 15, 5x, *NJ-‘,I§,SX,*IN-*,IS,SX, +*L(N K -* 150 Si §8RMAT : :' 'iox’ ’:§':’ 1%,: H5 33 %; x*MAx * .15 3 *MIN * 15 *EggMAT? 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IPLAG-O NAME-NONE (N DO 220 x- NAME IFL?G-I§LA¢+1 M-L N,K ...1§§¥§.91.§I£11???§11115111951159?§.???§§§2§. 1.89918199959111 88 188 8: '12 SE(I J)-o.0 CONTfNUE §QN:INY552(7 7 -B(N)*A(M)/LB(M) BE 21 --sz(i.1) ...1111.1?.9????.§§??.9?.¥11?1?.?1.§¥?¥?1?1....................... 88 818 8:1'18 32(I J)-SE(J,I) IQN¥§NI§ES Q GO O 601 NR§TETIE,§O§ I?§E(I,JT,J-1,12).I-I.12) CONTINUE TO TRA§§ on" 52(12 12) FROM LOCAL TO GLOBAL COORDINATE *fit§§£%g*** t£§OI§§**§$£§§§§§§**¥§z§£§*£§t§¥9§égti*************t** 11 1911"“) T m ggng§;§%,§b§ ‘ 253??,J?,J-1,12),I-1,12) ASSEMBLE STIFPNESS OF EACH TRUSS ELEMENT INTO .**§$§9$I§§§£i¥£§§é§t§I£§§§§§§ttt*********t***flit***************** EALL ASEMBLE(M) ONTINUE IFEISTRES .E .1 REWIND 19 IF IsTRa E 1 REWIND wNITE 4, 5 (s I.J),J-1,MBAND),1-1,NSIZB) IF(ICAL3.NE.O) GO TO 991 * ****** 332 * ****** 700‘ + non NM NE’ NE’ 1&93 §§£§I.9§.$§E.91§§9§é¥.£21§§¥§§§.¥é$§1§........1 WRI%g?6i:;3§??,S(I,1) c NTI 53 F MT . *plox'*1..plaplox'.5(1'1).*132101‘) £§£§I.993.9§.2§§.§I§99$9§¢2.§I£§E§§§§.¥é¥§£§.......*i NEITE(61.701) 3§§3§3§TART+ IF(JE .6; ANU)JEND-N3AND NEITE 61 2 JSTART,JEND no 71 I2- §§If£ g§§$§N3%,7 3 s Iz,a),J-JSTART,JEND) IE?3-§9izy,z.1,12) 831888 1 zHAA E3 Hfi*m> HHHU$MH a ZMmmH H INATE **************t "no §(”5)c S 9158(160 a) T3-1 2),I-1,12) MBLE STIFFNESS OF EACH ELEMENT INTO STRUCTURAL *Hm Hum * ***********tii*fitti***fi*******i**********************t FN *** EMBL LE(M) as; as Pi uiwnm Cung _- r I 3: :fifihwwyjfi va'nmcnvA ZH ’~‘Zf”§ %§S. )) REE§NB S(I,J§,J-1,MEAND),1-1,NSIZE) UfaqerHm ZAQHAA hw¥4H RETURN FORMAT(E21.15) END SUBROUTINE ASEMBLE(M) i**i9*fi***************t************it*t*****ii**********fit****i*t* Tgm PROCESS AND ASSEMELE ELEMENT STIFPN SS MATRICES AND NODAL ESTORS INTO THEIR QRRESPONDING TRUSTURE ARRAYS AAAAAA AAAAAAAAAAAAAA AAA AAAAAAAAA COMMON/1/NE,,NUMNP,NUMEG,mLE(54),NUMEL(3),IPAR,ICAL1,1CAL2,ICAL3, COMMON/Z/ §,§:QW BAND 1% ,x(36 30), 2(30) 16 WE:Q:P? NO§EI(54), ,NODEJ(54), A(54), 183(54), KT(54) ) 1 SP 123 36), 1037, IFLAG CTRUE msgxgsnass ARRAY AND LOAD VECTOR g********* t*******i********************t************** .1) GO TO 90 5128 AAAAAAAAAAAAAAAAAA \OQ*QV\: MTWHWDHZ ‘2‘; *m *OC AHNANVW u: Cy43 z». .+ L. Z~4um~>'t cllole mcwxmao «a: 2nd tn 3 ******************** 200 * A i A I) A A I/ A t t 3 A A I A A x I A t 5* t A II t * 31v * A IJ * t I. t T A t I t P A x x T A I A P5 A R SA N I A CD Rfi * 3 * SN 0* I o A BA If x I F. A .03 CA 5* A 5 WA I. m A F A 3J LA . O A IA ”AA J m... I I NA JE A A x AOA . D \l O. .5 36* I I LA 1 I AA ms I * JKS “It T ( LA JA I IDA U R. AA I I I A 30 I DA JXA XNA I I 0* s 8 II* as I NA .D R A T“ N A 3N T A m I “A It A E I A 2 I I GA T I AA KA X NA L * I35 SI* LR \I 5* IJ I 5m“ UW I 0 SA N3 E F. 1 EA 0 .A I 0 NLA O 0 0 NA 1.. I I I 3 FCA F 3 2 Pt 1 it 1 FN* OP I F! o 3 0 PS 3 0 II* O N 1 IA 0 0 O 1 2 N .2 O 4. T A W0 6 TA 0 I T I 1 .3K T I SEA 0 L 0 I\ § 9 I I I 1.1.! 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TIIITTW AA T IgITTT NNlKKlIITIIT . TI. TITIKKIIIT .TII I. I. I“ FOA JKKKK CO (A (A A (A AI(NN AN ( (NNN AA (( (A (A A (A AN (( (AI 3 (A A (A A A FIOFNOIOFI IFOO“ A o me5000 IJwFF PI PN OPIOIomFan-sa OIOFN OFJOJO A Far—rats... IIGINGIGII IRICC A C I CCC NN II II IN IGIIGIC II IJ F+.GIN JGIJGJC A IIIII 1 21.0 o 5 O 12 g 0 3 5 o 983 05 0 0 0 so 00 00 0 000 0 o 1 ll lfi; 2 9 2 3 33 . 4.4. l 2 3 34 56 7B CCC9 999 l l I I 11 IA A A l I ll ICCCC CCC * I ,SX,*JE-*,IS,SX 14/I / (IE, 2 sx,£1é $-*,821 ,1§ ,JE 2([1 59> IE 11%? S(II X. 5 X *‘k**********i*********i’i’tiit‘ktt‘ti’tt********************** * t t t t t .3 t t t I v * ... fl... 3 t r ._ ... . ... IWH C \1 t o a c... I 8 t D t 50.. p 6 * 33388833 A c a m... 2 1 ... 22222222 0 t . t u l\ * IIIIIIII L a Fs.‘ C * 55555555 9 . t C R ... NNNNNNNN O * OF. I I * I I I I I I I I T t 0.. r ) ... 1111111 N ... . t 1 0 t n u a n s a n a I t DR... L 1 * IIIIIIII D "om" a 8' m S" I I I I I I I I E 6 a TC... I 6 Z S... WDDWDDDD 5 . t W: p 1 I E... NN WNNN u .. "m n m u m. m. Ammmsmmm l ) C F ... I”: P G p. P... E ”MMMMMMM m I * ”on I I a“. fl“ 2 I ILLLL I I I O I q! I 11 1‘1; P ) t 0L: ) .56 I 5.. S KKKKKKKK H ... C t 3 L6 p t N u u u n u a a u 5 2 ... CE... ( N F1 ) Rt ' JJJJJJJJ m I * As* L E Qt I a“ l IIIIIIII 2 a N... G .3 ( a 0 I t SE: I TG R N: I JJJJJJJJ L I *SD* EEI IT; I IIIIIIII r * EN. I D\I I Lt ) IIIIIIII L H * "0* r r 7.8 I.\ t 2 A 1 ...FC... ) D 6 E... R 55555 D I * F * 4 m )1 )\II * I N I ... TT* ( a 32 56 0 K I * s * E ) .6 00 C. 3 I \II)\-I\I)\I\I) 0 ( ... 0.. L .88 r 22 U* 0 J 71234560 1 P ... E5: . W66) p y m... 2 p 22222222 5 1 A). .. L... G 118 11 .. 0 p ) 0000000 1 \I /\I .. WA... 0((6 66 5... 91 J 22222222 1 OD L/H * .. CRPl (( t 6 . ...... . . TN 0 Axld. .. C .... N I BE D... 0( 021 0 1111111 0 0 A 0 II o ... 03... c) 1 TT 3... TE 6K( 6 666666 T 108 1 T(3 N .. RIr : 0.5)G II S... T .5 16M IRI m a T .. D. E 6 v R N... I 01.7\ m 135383383 0 r O N p * SD. 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F123? 3123....0 FFFFFFFFO FOLKO ...»S\FF OJ F+F S t * ccflccccnhuII * IIKKKI WKKKIG IIIIIIIIC T.DLKDJIIDDJI 0 0 0 1 0 0 0 0 0 c C2 2C CCCCCCC CCCCCC CCC 5 6 9C 202 .. .. t .. ... ... ... a .. ... t .. * ... .. .. ... t t * ... x) .a t t \I ... a a \r 5 .. ... .. xI Ix .w ... t S a .. §§8 .. Ix .. .. EEEEEEEE .. .. ... NN NNNNN a E .I * I I I I I I I I * R .. ... t C .. .. a o a u a u a a a U i * IIIIIIII t n * . I I I I I I I I U .. ... t R S .. ... xI DDmmD D T t a w NN N N t s F) . . ... Anna»... % ... on .. a . MMMMMWMM fl... 0 S) / * * l ) IIIIIIII * SI / .. a a Q 11111111 5..) S E( x! ... t I n KKKKKKKK ... Q 5 NR I t t I c u u u a a a n F... E E F I\ ... ... x) o JJJJJJJJ O... N N\) FR R * * I 1 I I I I I I I I * I F4 1% ... .. ( u 5.. 1 PI T R .. .. R I JJJJJJJJ 5.. a I p 5C m * t I I I I I I I I I I a. I TH W t t I \l IIIIIIII N! I 56 R c ... ... Ix 2 P... x) U A W .. .. K 5555555 F... R0 E”) x) ’ . I I 1* AR N 6 6 W * * 00 \II 1 I T’ E” IO 0” o .. t 73 0 K 5.. m N LL6 6 D A* * 8 o I I * In“ 10.]. ... T.. .. 22 0 J 71234560 R: I L7 DDFLF ) A...0 N: I I 2 : 22222222 .A..1 : E... I o I Dr 50 0.. 115 9 xi 00000000 We. - E4 55 a a I ... 15 I... 6681 J 22222 .. J I lewxl 0 I N* 1 II: 0 I o I IIIIIII 1* I s I EHSH 0K 0.. m Mi ”:3 Ix 82I 3 WIIII L.. ) N5 DZN2 1K I.. m .. .Tmmm 1KIx 1 66666666 ... J E N .E I Ix To 8.. II .5 an I W 00202 N 5 A: nOvO N.. “W I m IEmEfiEEE I CINI L . 5.. m 8.. 0 K( 1TTTTTT S..Ix 0L.......5N .0. 0 \I N* G I 6* G KIIIIIIII Ni 5 86666616 .U CI\ 5 OJ 8... ) 0+ ) .. xix!) 0)) m )WWWWWWWW 8.. Ix )HC11111216HHRHR N GJ mt 0x1C BJ A: 000) G50 5 mt Ix 53HEEEEE33158H6H I A o ... .2N M r T... . . .2 12 1 .. 14782345784222 L BIL 0* Q o to k A* 8.8 O ‘0)8 1“, QIII o. \l I IIIIIIIIIIIII I I M0Ix C... 831C175) D... EENQ K722 K7 201234.567 C: 0 1.. .. ... ... t .. .. ... .. .. ... ... .. .. ... 3 +5 ...NaN-SK .....E ..pr .0'........ ..1 21.00000000111 N ”AB-Em E “M..m.I.J-Ix "9.111. Ell 036100000000... at .4 Btuattgttwtt..**t I KTTIU U ... R E )6. Ct LLLR L§6NLIGEEEEEEEEU .. 4 m . .JNN N O... OAOZOJ. E... AAAA A .(Ix+ IA . Ix . . . . . . . .N 0.:(m TTTTTTTTTTTTTTTT IJIIII T... CP4I3 (II Ht CCCP B3BEl2B3OE3333§333I T... E A O I ITTT 5* NIISsIII Ct IIIII‘OMKTTKKMKTTKKKKKKKK 5* TI m R quLNNNN ..(Ix N (Ix .. u .. 7x11: 3 7x .. IW W B IFIxOOOO .. FFW - CCC .. FFFP123FWW123FO FFFFFFFFO 1 “E n OOOOOOOOOOOOOOOO U IISCCCC .. II K SRC t IIIIKKKI KKKIG IIIIIIIIC .. R FFFFFFFFFFFFFFFP E S 0.391239536700000 002... 000 0 0 5 0122222235678 0111 34.5 6 8 8 00008080000000 1111CCC 111CCC 1 1 ICCC C I22222222222222C CCCCCCC C 2([3 tififitfittt...fiittfiflfifiifttittftflfiflfit..fltttttfitif**.fi***t**it*ttt*t*fi LVE SYSTEM 0? LINEAR E UATIONS S*D-R BY CALLING THE TgPIOPRIATETSUBEOUTINE Q ms DL0NEARS STIPFNESS 00000000000 D: VECTOggF AT §A835;88§M1Nm ION EEQUATIégx :OLIN éééngEBD” Pow QQCS9§E§PI§Qttttégggl 0:39" 9..II“ titaffizggigttttttttttt ‘z§¥M0NS§1/N2fi,NUMNP, NUHEG. LE(54) .NUMEL(3).IPAR,ICAL1,ICAL2,ICAL3, MMON 3%??? g?” géii??i"?3§?£§s§§§§‘?w am. 10) mm, ?H IGAU NTEGE‘ PéIgégN IF( GA S.EQA1) GO -.iitigmeéméa ‘3in9????9?f§§§§3?§i?§§§1§§§3§é§1¥?§9§§....... 10 BIIIIIII'l'NEQ . ttisfigstt9¢z§ .§§”ERAI£9.*:9§ C§9nglggiggfigggézggg§i...Q.*’**..*.*. ????:. fl??? magma: Wm“ SOLVE SYSTEM 0? ‘NEQ‘ LINEAR EQUATIONS QO*ggggéggfigggggzzggfiggtgézglg*£§é9§§i§§£¥£§52£9§19............Cfifl a??? ’r§.2%§8m . . : (N. )/S(N.l) 7 0 KOL,H3AND 750 é'ftag- (I.J)-C*S(N.K) ég’t'fg'fié 33 m 7 7 g ...:9????9.?§299???? .95.99§§2...§.1§é¥§§.§§£¥£§¢2£9§1............. DO N- figégL “I M820D) GO TO 820 éI)-g§é)-SIN L)*D(N) 3%8 S NI—D N)/S(N, 1) wag§9&v§*: 93 *ggggggygtgzfigésgfi§gggzgzggiegtttittiit*Qtt**ttt***tit DO 860 M-Z ,NEQ N-N; W1 I? E?NL LI” EQ. 0. )GO TO 850 K-N+ D(N)-D N)'S(N, L)*D(K) EONTINUE ONT IF(I 0000P man 0 W000 '0 828 m as GAUS. SQ 1) GO TO 140 CHECK DATA GENERATION 00 p C) CKND NW43 CKXD 000000 000an 0NNNOH ND C) 883 0000M nnnnnm 0M4 +L CW 2(L4 iii.itfittittifitttfiitfltffiiitttttttiitittt**********'*i**9***tfittfltfi ;( RIO? ) ICALZ-O gig????§?:°;:::1,m, ronnar * I? LINEAR SOLUTION//) Pamuwfi: * :??>§ 3‘52??? £3?“ r EMA? 6 xuzan SOLUTION//) END SUBROUTINE RECOVER tttfittttttit:ttfifiitfitttttttttttitti.titttitt*tttctittittflttttttttt MAL D. O F '5 OF THE STRUCTURE AFTER tiirgéylgtittit§fi Yszgniegit 299;?Inéifittfiitifl.**.***O*QOOCOOOCIOQQ*OO COMMON él/NE, NUMNP. NUMEG. LE(54) NUMEL(3).IPAR.ICAL1 ICALZ. ICAL3. :§:%§7§6N§fo$565381(?$2§?:"3?§26§5é§7§53),c4(1ea,1o).RC(1sa). %fii§é PRIOPTN 1N12& NIOPTN E“ “PRIOPTN. wzgéégoI SOLEZS 13 g-NCN éééfifl??? NI??? 9 é?§C(J)-DUM)/SC(J.M3AND) ?§i:§568¥ 6° T° 12° mug? u82 I~I C; I :+¢m40c). uz H (”alkww4uIZ: zgw anUHH 'hfil a<¥§ n ( 1,2010) (I,D(I).I-N,NSIZE) Eafiaz 5?? ?? $82uA$ 11: :fim “i“?§% -?t35f1é) S SUBROUTINE IDENT *it;8§;t*****fi**fi**it*fi**t*fl**§i***Oiflfiifl*i********§**t;*********t MY THE DI ISPnggMENg FO UND INm SOL$T% *aa389駣9N§*§: :Qtéggt tiaggtiitggugttitt* .§§§ tttztt§99§s§****t cg¥gg§é él/NE NUMNP,NUMEG,LE(S4),NUMEL(3), IPAR,ICAL1,ICAL2,ICAL3. HMO oN/2/N fiQ 8 ND EI §?§ mg??? 32???? 4;??? 52?"??? a??? if??? mum) m“ D 62(168L 63(168), 64(168, 10), RC(168), 2205 $§g§ “(30, 6), V(30. 6) (1 a, ) /9 EM 123%)???" tittgggzig£ :19! OQEQQ£§EEQS§¥§§yéttItittittttittttttit...tttttttt 15{PR§38§T§93831) I$QIZ1° 2000) (m1t§n$’.'¥bm.3) 00 1'0 240 sun: I? N 310 om 9% man .8528 MESS .. 33:8 W 12?:c 2 a .0) WRITE(61,2010) 5 11:15.1): § 8531' '§p-N1 {g °5§3é upon: r(1a?np.1§§ 150 155 150 15° 5319121)”; ég<1 3850. &0) wnxrz(51. 2020) u.»p,1,w(up,1) 55 §0(I I 3038' 0) waxwz<51,2020) u,»p,1.w(np.x) 160 "a. 1? 501;).11.0) 00 20 170 § 50 17° 8'3--1§s¢11m §§(I 3630. .g) HRITB(61. 2020) M.NP,I,N(NP,I) 133 A un,1 )“Bgo. .200, 210 w ' “1’ ffi’g‘fi . ($81 33% 0.0) wnxr:(s1.2020) u.up.1,w(up,1) N N I I 20° éo‘ggiéi 23. 0) waxrz(51 2020) n.np,1,w(np.1) 6 210 gifig IEEDW :@ (I A 2 .O) NRITE(61, 2020) N.NP,I,W(NP,I) ammo QumJ cxxa Mann-0 (3000 Efififi I Hgég; , gaff? fi§:1gwmm§ 95:5uzxrs on sacs anznzxr) - * 523 ,2. §,,12,1a,,1L 2a)-,525. 15) SUBROUTINE DECK(EIGEN, IDATA, ICAL‘) Stififitittttitiittittflifi9*it;it.fl.;*.***.*.Oflfl.****;t*;.*Qtttfitfiififififittit * Em ROUTINE DECK INCLUDE THEI EFFECT OF sD NTHB ARCH * : BY! 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SOL sag” Amy‘s é%?’//13,ZHKI,I3//) 21% FORMAT * *Iafge ) ' ' 2 FORMAT * *,E 6 ii 1 :8EEII : :'3215- 2 § FORMAT * *Igal I gizg §8RMA$ : :' glg' 2126 FORMAT * *:9§i6: 31$ ggkflA; :6:,30§vaggog Y(I), SENT TO GAUSSOL As XB(I)//) 5 gggéag :9::§§,isgfiggéggggUgngf%}§EIGENVBCTOR,SX,6HILAST-.13) 2 g FORMAT *-*i§31.§4z ' g FORMAT * *, x,E1 .8) END C c c C c 00000 00 nanny—a own »«3 00000HH +44 bh. um: 0kw4N5 FWVNWFWJ 0cm) can own PK) cm: cxn CK) 00000 0 NNHO F +4X,6HNO 2137 SUBROUTINE ENDFORC tittittitiittififitttfitttt'ttitt‘ki‘i‘kii‘iiti‘ii**********fi*************t T 0 NT END F «ROORRQRS*****§*§&§M§****ttt*9*9* *ttttttttttti*flttttttttttttttttt ggggggél/NE, ,NUMNP, NUMEG LE(54) ,NUMEL(3),IFAR,ICAL1,ICAL2,ICAL3, COMMON 2 N5 N0 MEAND EN Eggfigggggégz 32, EEQIEYE 0, 6) E(30)% 30) z<30) C?MMgN{S/E( 2% PNQJ (54) NODEJ(54) A(54) 123(54), KT(54) FORCE) 22%.. 1%))???“ COHMON/§{7DNT1 2h ”7 v)” COMMON/Iz/ULOC RCOL(9),M5UOFTN, NIOOFTN wRITE 61 2000) DO 3% LN-l NUMNP DO I-},8 PINT LN,I - . ...2599§§§.§Y§§¥ .§&§¥§§I.QE E095 §9§¥§§2.§§ 992.................... DC 200 K-l N IF(NUMEL(K2 U§8§0) GO TO 200 NAME-NUMEL N) no I90 K§-1,NAME M-L R xx NI-NOfiEIiM; ¥€T§°DSJ M so 0 so ggAgé§ g5) (TSE}I,J),J-1,12),1-1,12) REAO§10 i0) ((SE(I,J),J-I,12).1-1.12) CONT NUE QBIAIN RE§9LTANT m; *t* a ****t* *****R 9*: *ttitit*tttttttitfittitttttttttttttttttttt 1 0 DN(I?-DN(H E( OC M J) PéNTJLN, I)-F1m {LN, IVLDN 0 I-7, 12 ON DO $51 J- 12 ON -ON(I)+SE(I J)*ULOC(M, J) 80. Ho—DH HZ ( CONTINUE fittttirgtn *§¥L3§§Ig§9§9§*9§*I§§.§99§§*9§.§§§§.§§§¥§§I*A*E....ttgg. WRITE(61,2010) M,(DN(I).I-l,6),(DN(I),I-7,12) CONTINUE CONTINUE REV IND REWIND 10 RE RN NO 5 ON EACHE 6HNODE- 1,6 W3;//)15. T321 103 t.* DE- ”‘0 END 2113 SUBROUTINE NLEIGNP(SCALE) *******t**t*********t*ttt***t*****************t******i************ THIS ROUTINE WILL COM’UTE THE EIGENVALUE OF THE UADRATIC EIGENVALUE PROBLEM (K+L N1+L*L*N2)*X¢0 ....¥$.9§§§.fl§.§99£§£§9.§§§9¥é.§§&§£.53? .99.................n.... EXTERNAL DET REAL L 0000000 0 0 XTOL FTOL,NTOL READ( 0 10 , ,DINCR ; XTOL,FTOL,NTOL,DINCR WRITE g},% 100 ohoo~e GD ‘1” ,NE-i OV‘ 008“) O”. \ V V“ F A A GO TO 110 HOD?! HO HZV 0mm) oon ~49, UW%D 110 U 3H marvrwz g EgET,eOB+ng%OFTOL,NTOL,IFLAG,SCALE) lemrIIOOIm -¢moc '11:) Nflkwinfi 00w 8m owl 2. é L,ERROR,FL cflfl) ~»\ mvwfln\£mw1 m C C) CGHUOyHfiI C E2 ”HUN” 0"! aemexee ZZAfllwb M4“ M 00 \*O' \N is’EZS.IS,IOX,IZH PLUS/MINUS ,Ezs.15// / - ngEEEEEPEEEEEEEIEWL .rlo.7/// 3,1 HDETERMINANT//) I 3 MW §8§WMFH0>U00>H337§ 0 CK) wh)|4 cm) OC>¢D ax: HMMNNH- Elba wuxxn :3 CK) \IFPU\W \OI*W\H \w i V h—I\o SUBROUTINE MRGFLS(F,A,B,XTOL,FTOL,NTOL,IFLAG.SCALE) *ttttt*tifitttt*fi.***t**i***it***********************t*t****i****** ITERATES TO A SUFFICIENTLY SMALL VALUE OF THE DETERMINANT OR TO A SUFFICIENTLY SMALL INTERVAL WHERE THE ROOTS MAY FO ND tittttttgttttt*ttttttiit.*tttt*tttttttttit*tttttttttttttttitttttit IFLA -0 FA-F A,SC LE SIGN A-FA AB (FA) FB-F(B,SC LE 0000000 0000000 0NN N NHO ECK FOR SIGN CHANGE iii 9* it! ** ****** titttiittttttt*ttttttttttttit*tttttttitttt IF(SIGNFA*FB.LE.0.) GO TO 100 IFLAG-3 WRITE(61,2010) A,B RETURN 00 w-A FW-FA DO 400 N-l,NTOL CHECK FOR SUP ICIENTLY ***** t F ****************** IF(ABS(B'A)/2..LE.XTOL) RETURN CHECK FOR SUFFICIENTLY SMALL DETERMINANT VALUE PROTYPE83 FOR INCREMENTAL LOADING IN MOVING COORDINATES 0000 040 SMALL INTERVAL ************** *t**t******t*****t****tttt 000 0000 2119 iFéABS(FW).GT.FTOL) GO TO 200 B-W IFLAG-l RE “fans 200 FA*B- FB*A)/ (FA PREVFw-Fw/AB “PW Ew-E RMAT ////13fi no CONVERGENCE IN, ,1gnp I T / (D + CC! um) H CXD‘O TERATION END FUNCTION DET1(SCALE) ***********i******.*********************************fi*fl*********** THIS FUNCTION COMPUEES THE VALUE OF THE DETERMINANT 0F MTRI X -K+1+N *itt T*§*m H***t ****i***fi**fi*******i******flii*********9************ common 2 N I ,NE ,NC MD MBA n IEIGEN counon79¢ s?16 5? p?168, 36 ,iDET IFLAG T T 000000 0000000 0NN N0 0 M i "00")", H R #8. hr firtho - H $2 um I I Flt!) 99$£9§.9§.¥§I§£§£§§9§§.§%§¥£§é3£9§l...............u , 8 ND NLLL .EQ.0.) GO TO 380 ,LL)/S(LN,1) ItUfi++iJD :zgrmxanm~ END c/// SUBROUTINE GRAPH(LB) INTEGER VIEW WICHBIG LOGICAL EIRs T COMgggél/NE, ,NUMNP, NUMEG, LE(54), NUMEL(B) IPAR,ICAL1,ICAL2,ICAL3, 5 com ON/2/Ns Q N OND NEAND CBMM8N/3 3i63§?¢§ 12856&§?l 28§? w: ?166§, %9&NVTR(168,10),RC(168), gems AU? DATA EIRST /.TRUE./ RENIND 21 IER (. NO ) vOOTO 100 DJééfiV aw READ 681 NICNEIG ALSE. 100 CONTINUE 10 20 30 40 Zill WRITE 21,1 VIEW WRITE 21,1 NUMNP WRITE 2 ,1 N88 WRITE 21,1 WI HEIG FORMAT (15) wRITE (21 29) ((IA(I.J),J-1.6).I-1,NUNNR) EORMAT 613 ¥8%EET(%%éggzlé§(I),Y(I),2(I),I-1,NUMNP) NRITE (21 401 £(EIGNVTR(I,J),I-l,NEQ),J-1,WICHEIG) EORNAT E21. 4 RETURN END